The victory of Donald J. Trump will test the ability of America’s European allies to maintain solidarity, do more to build up their own militaries and defend their economic interests.

In anticipation of a Trump victory, there have already been efforts to try to ensure continued support for Ukraine, continuity in NATO and to craft a response should Mr. Trump make good on his threat to apply blanket tariffs on goods imported into the United States.

But the Europeans have a long way to go. A second Trump presidency could serve as a catalyst for Europe to fortify itself in the face of a more undependable America. But it is far from clear the continent is prepared to seize that moment.

With both the French and German governments weakened by domestic politics, a strong European response may be difficult to construct. And even after one term of Mr. Trump and a war in Ukraine, Europeans have been slow to change.

“A Trump victory is very painful for Europeans, as it confronts them with a question they’ve tried hard to hide from: ‘How do we deal with a United States that sees us more as a competitor and a nuisance than a friend to work with?’” said Georgina Wright, deputy director for International Studies at the Institut Montaigne in Paris. “It should unite Europe, but that does not mean Europe necessarily will unite.”

The unpredictability of Mr. Trump — emboldened and empowered by what may be a Republican sweep of both houses of Congress — concerns European allies, since unpredictability cannot be prepared for.

But they also know that Mr. Trump will maintain some clear positions. Those include skepticism for multilateral alliances, an admiration for President Vladimir V. Putin of Russia, and dislike of President Volodymyr Zelensky of Ukraine, said François Heisbourg, a French defense analyst.

The Europeans will continue and intensify their efforts to keep lines of communication open to a new Trump administration and the key officials within it, even as they maintain close ties to American legislators who support the trans-Atlantic alliance and NATO.

The main issues are the economy, security and democracy.

When in comes to the economy, the European Union has been planning for months how it might deal with a President Trump.

E.U. officials have put together an initial offer to buy more American goods to try to forestall new tariffs, and drafted reciprocal tariffs on American goods to respond if Mr. Trump does go more protectionist.

On security, there are worries about what a Trump presidency will mean for Ukraine, a war Mr. Trump insists he can end very quickly, and about Mr. Trump’s intermittent threats to withdraw the United States from NATO.

Mr. Trump has been correct and effective in demanding more military spending from Europeans, said Mr. Heisbourg. “But NATO’s Article 5,” a commitment to collective defense, “is not supposed to be a protection racket,” he said. “But that’s Trump’s position, and this time he’ll have more power than he had in the first term.”

Article 5 depends on credibility. Some, like Ivo Daalder, a former American ambassador to NATO, think Mr. Trump could destroy that credibility and tempt Mr. Putin to test NATO simply by saying that he would not defend any country that does not pay at least NATO’s goal of 2 percent of gross domestic product toward defense.

Currently 23 of 32 member states do pay that amount or more, including those states most vulnerable to Russia, like Poland and the Baltic nations. But there is also general understanding that 2 percent “must be a floor, not a ceiling,” as NATO leaders keep saying, and that countries must spend even more given the Russian threat.

The new NATO secretary general, Mark Rutte, a former prime minister in the Netherlands, knows Mr. Trump from his first term, and Mr. Trump has praised him. Mr. Rutte has told Europeans that they must spend more in their own interests, regardless of who the American president is.

At the same time, there have been some efforts to “Trump proof” support for Ukraine.

NATO is taking over the Ukraine Contact Group, which coordinates support for Ukraine, from the United States. NATO countries have promised to deliver at least 40 billion euros, or about $43 billion, to Ukraine next year, the same amount as this one. And the Group of 7 nations have agreed on using billions of dollars in frozen Russian assets to provide Ukraine $50 billion for next year.

Poland and other countries of Central Europe, including the Baltic nations and Hungary, had a good relationship with Mr. Trump during his first term.

The foreign minister of Poland, Radoslaw Sikorski, said in Warsaw that he was in regular contact with security advisers around Mr. Trump. But Europe, he said, “urgently needs to take more responsibility for its security with increased defense spending.”

He vowed that “Poland will be a leader in strengthening Europe’s resilience.”

That would be best done in cooperation with Britain, France and Germany, said Jana Puglierin of the European Council on Foreign Relations in Berlin. But like Mr. Heisbourg, she said that the weakness of the French and German governments can undermine that goal, and that Europeans may instead try to make bilateral deals with Mr. Trump, as they did last time.

“There is little leadership in Europe, and Europe can’t be led by the Commission or by the European Union institutions,” she said, referring to the bloc’s bureaucracy in Brussels, “but only by its strongest members.”

But Chancellor Olaf Scholz of Germany chose to prioritize his close relationship with President Biden and has not invested in Europe.

Paralyzed and divided, the governing coalition in Berlin collapsed Wednesday night. “Germany is seen as a problem in Europe now,” she said.

Most importantly, Ms. Puglierin said, “We in Europe must confront a lifetime illusion, thinking that Trump was the real aberration and overlooking the deep structural changes in America,” including the shift toward Asia and a growing fatigue with its global responsibilities. “So this is an election that Europeans should take very seriously,” she said.

The German government’s trans-Atlantic coordinator, Michael Link, said Trump’s re-election meant that both the European Union and the European pillar of NATO had to be strengthened and avoid divisions.

“We can’t just passively wait for what Trump will do, or what Putin will do,” he told German radio. But he also said that Europeans must “make clear what we expect of the U.S., that it must fulfill its NATO obligations, and that if it disengages from Ukraine, in the end that would only help China. That if Russia wins in Ukraine, China wins, too.”

There is also concern about democratic values and the rule of law, and Mr. Trump’s evident admiration for those he considers strong leaders, like Mr. Putin, Prime Minister Viktor Orban of Hungary, Prime Minister Benjamin Netanyahu of Israel and President Xi Jinping of China.

Mr. Trump is seen as the standard-bearer for those populist center-right and right-wing leaders in Europe like Mr. Orban, who has established what he calls an “illiberal democracy,” as well as Prime Minister Robert Fico of Slovakia and Prime Minister Giorgia Meloni of Italy.

His victory is inevitably going to inspire them and encourage others to duplicate more nationalist and less liberal policies built on stopping unwanted migration and on protectionism.

Europe is already seeing a decline in support for democratic, liberal, progressive values and the rise of extremist parties on the right. Mr. Trump’s victory will embolden them and weaken Europe’s coherence and its voice.

“Spreading liberal values is a lot harder when the president of the largest democracy, the United States, openly contests them,” Ms. Wright said.

For the past year, the United States and China have tried to manage their rivalry to reassure the world that tensions between the superpowers would not spiral into conflict. The return of President Donald J. Trump to the White House threatens to upend that delicate balance.

As a statesman, Mr. Trump’s calling card is his unpredictability. He revels in mixing threats with flattery to keep his counterparts guessing. On China, he has vowed to impose blanket tariffs on Chinese exports, and threatened duties as high as 200 percent if China were to ever “go into Taiwan,” the self-governed island claimed by Beijing.

But Mr. Trump has also praised Xi Jinping, China’s top leader, saying on a Joe Rogan podcast that he was a “brilliant guy” for controlling 1.4 billion people with an “iron fist.”

Regardless of the direction of Mr. Trump’s rhetoric, Beijing has likely concluded after Mr. Trump’s first presidency that he intends to wage a fierce rivalry with China, no matter what he says.

“Xi Jinping is an unsentimental leader with a dark interpretation of America’s intentions toward China,” said Ryan Hass, the director of the John L. Thornton China Center at the Brookings Institution. “He would be open to a friendlier leader-level relationship with Trump, but he would not expect a warmer personal relationship with Trump to dampen America’s competitive impulses toward China.”

Buttressing Beijing’s view is the fact there is bipartisan consensus in the United States about confronting China. Trump may have started an era of bare-knuckled competition with the trade war and increased American support for Taiwan, but that approach didn’t change under President Biden.

If anything, Beijing says U.S. pressure has only intensified. Mr. Xi has accused the Biden administration of unfairly containing and suppressing China. He points to the deepening security arrangements between the United States and its allies and partners in Asia; restrictions on Chinese access to American technology like advanced chips; and the use of U.S. sanctions to punish Beijing for its tacit support for Russia’s war in Ukraine.

While the precise details of Mr. Trump’s foreign policy agenda will likely remain unclear until he picks his cabinet, China is already showing it is more prepared for whatever is in store compared to Mr. Trump’s first presidency.

In just the past month, China has been cozying up to American allies and partners who might feel uncertain about the future of Washington’s reliability. It struck a deal with India to ease its border tensions, and Chinese troops exchanged sweets with Indian soldiers during the festival of Divali, along the disputed territory. It hosted senior British and Japanese officials in Beijing to smoothen ties. And it lifted restrictions on key Australian exports to China, like wine and lobster.

Over the years, China has also doubled down on efforts to become more self-reliant on technology, investing billions into developing its own top-of-the-line chips. And China has continued to build up its military. Mr. Xi, in a show of strength earlier this week, inspected his country’s elite Airborne Corps, paratroopers trained to “liberate Taiwan.”

China’s bid to insulate itself from a potential Trump shock, however, could be constrained by its weak economy, which has been battered by a property crisis. China was not nearly as vulnerable during the first Trump administration, and it may have fewer options to retaliate in a trade war.

Some voices in China are urging the country to exercise restraint. Jia Qingguo, a professor of international relations at Peking University, urged China to prepare for greater competition with the United States not only by investing in its military and economy, but also by avoiding accidental military conflict in the South China Sea and Taiwan and sidestepping unnecessary disputes with other countries.

But some Chinese analysts like Zhou Bo, a retired colonel in the People’s Liberation Army and a senior fellow at Tsinghua University’s Center for International Security and Strategy in Beijing, said that China was getting better at standing up to the United States because it has weathered the opposing styles of the first Trump presidency and the Biden administration.

China responded to Mr. Trump’s blustery Twitter diplomacy by introducing its own brand of muscular and acerbic statecraft known as “Wolf Warrior,” a nickname inspired by ultranationalistic Chinese action movies of the same name. And to counter President Biden’s democratic alliance building, China aggressively courted deeper ties with developing nations and with Russia. As the United States has built ties with Taiwan, China has ramped up exercises near it, including large-scale drills to encircle the island in a simulated blockade.

“Some people in China say Trump bashed China with a hammer and Biden cut China with a surgical knife,” Mr. Zhou said. “We have experienced both of them. But the trend is, China is gaining in strength, despite the stress.”

Ties had sunk to their lowest point in decades in early 2023 after the United States shot down a Chinese surveillance balloon as it floated over the United States. But the relationship had stabilized in the past year as the Biden administration has emphasized intensive diplomacy — dispatching the White House national security adviser, Jake Sullivan, to meet with China’s top diplomat, Wang Yi, several times.

Whether such engagement will continue under the new Trump administration will depend in part on whom Mr. Trump selects as his advisers. Those could include China hawks, such as Robert E. Lighthizer, the former U.S. trade representative. Depending on who is picked, his cabinet members might also restrain Mr. Trump’s transactional tendencies and instead advocate for a more ideological approach to China based on an opposition to the Chinese Communist Party’s authoritarian rule.

Parts of Mr. Trump’s agenda may turn out to be favorable to China. During his first term, Mr. Trump showed little interest in human rights, favoring trade and business deals first. In 2020, he told Axios that he shelved a plan to punish Chinese officials and entities linked to the internment of Uyghurs to avoid jeopardizing trade talks.

Mr. Trump’s isolationist-leaning “America First” policy could also lead to Washington weakening its alliances around the world. That could give China an opportunity to fill the void and expand its global influence.

It remains to be seen how China will negotiate with Mr. Trump the second time around. With President Biden, China sought leverage by agreeing to work together on fentanyl and allowing members of its military to hold talks with American counterparts. It is unclear if Mr. Trump would value any of those concessions.

On Wednesday, as Mr. Xi conveyed his congratulations to Mr. Trump on his electoral victory, he emphasized Beijing’s argument that confrontation would hurt both countries. Mr. Xi has sought to push back on efforts by the United States to define the relationship primarily by competition, seeing it as cover for a campaign to block China’s rise.

He said he hoped the leaders could “find a correct way for China and the United States to get along in the new era.”

Some Chinese scholars urged Beijing to move quickly to set up a meeting between Mr. Xi and Mr. Trump once he assumes office, noting that direct communication would be needed to manage differences.

Wu Xinbo, the dean of the Institute of International Studies at Fudan University in Shanghai, noted that during Mr. Trump’s first term, Chinese officials underestimated the American leader, possibly because they were unfamiliar with his approach, but that they should be more prepared for his second term.

“This means being ready for negotiations as well as confrontations; both will be necessary, and we may need to engage in talks and conflicts simultaneously,” Mr. Wu said.

Li You contributed research from Beijing and Hari Kumar from New Delhi.

As Russian troops march relentlessly forward with fierce assaults in Ukraine’s east, Moscow is unleashing a different form of terror on civilians in towns and cities: a wave of long-range drone strikes that has little precedent in the 32-month-old war.

Over the past two months, there was only one night when Russia did not launch swarms of drones packed with explosives at targets far from the front, including near-nightly attacks aimed at Kyiv, the capital.

In October, the Ukrainian military said it tracked a record 2,023 unmanned aircraft against civilian and military targets, with the vast majority shot down or disabled by electronic warfare systems.

Night after night, the explosions echo across Kyiv, with tracer fire lighting up the sky as spotlights search for the triangle-shape drones flying over residential neighborhoods.

Shots rang out once again before dawn on Thursday as air defense teams armed with heavy machine guns opened fire on drones flying over the heart of the capital. Debris rained down over businesses and apartment buildings, sparking several fires.

Though air-defense teams have limited the casualties in Kyiv — one 14-year-old girl was killed in October and more than 20 people injured, officials said — the Russians continue to unleash punishing bombardments with drones, bombs and missiles on other towns and cities across the country.

“The constant terrorist attacks on Ukrainian cities prove that the pressure on Russia and its accomplices is insufficient,” President Volodymyr Zelensky of Ukraine said on Sunday, reiterating his pleas to the Biden administration to loosen restrictions on the use of Western weapons to hit targets deep inside Russia. He also called for tougher sanctions that would prevent Russia from importing critical components for its drone and missile production.

“Ukraine deserves the same strong security as all our partners in the free world,” he said.

As Kyiv continues to plead with its allies to provide more sophisticated air-defense systems, scores of air-defense teams using heavy guns and other weapons form a chain stretching to the Russian border to combat drones as they fly in from Russia.

“It’s like bees swarming from a hive in spring to gather honey,” Senior Pvt. Yurii, 37, the leader of mobile air-defense team with the 27th Brigade of National Guard, said at the end of a 16-hour shift. Russia had directed 96 drones at targets across the country during that time.

“If we make visual or acoustic contact, we open fire,” he said, providing only his first name according to military protocol. “We use as much ammunition as we have. If we can’t handle it alone, we call for backup and another unit joins us.”

In recent weeks, he and other soldiers said, the Russians have been flying drones low to evade radar detection, frequently changing course to confuse air-defense teams, using decoy drones with no warheads to overwhelm defenses and sending surveillance drones along with strike drones to gather intelligence.

It takes about 50 rounds fired from the team’s Turkish-made Browning machine gun to take down an Iranian-style Shaheed drone, he said.

In a nearby field, golden ears of corn torn from their husks lay scattered among the broken and blackened stalks and the debris from a drone his team brought down in early October — a testament to the drones’ destructive power even when they are shot out of the sky.

A woman who gave only her first name, Khrystyna, and who lives on the 15th floor of an apartment building in downtown Kyiv, said she heard the distinctive whir of a strike drone drawing closer before dawn one recent day, like a moped speeding in her direction.

“We started getting ready to run out, and within literally five to 10 seconds, there was an explosion,” she said.

She had managed to get to an interior room before the drone crashed into her home, she said, most likely saving her life.

“My room was completely burned out,” she said, as she looked up at the charred remains of her home.

Andriy Kovalenko, a senior government official focused on Russian disinformation operations, said Moscow was seeking to use “around the clock” drone attacks to exhaust Ukraine’s air-defense systems, as well as to collect intelligence and “exert constant psychological pressure on the population in order to break the motivation to resist.”

The drone strikes are also most likely a prelude to the kind of large-scale saturation attacks using missiles and drones that have been a feature of the war, according to Ukrainian officials.

The stepped-up drone assaults are part of a deadly and unrelenting campaign that has been playing out for years as the Russians continue to try to batter Ukrainians into submission. The Kremlin is using not just drones, but also nearly every conventional weapon in its arsenal to hit both military and civilian targets.

In Kharkiv, Ukraine’s second-largest city, 14 people were injured on Sunday after a 1,000-pound Russian bomb slammed into a supermarket, according to Ukrainian officials. Roughly 380 buildings in Kharkiv were damaged in Russian attacks in October, Mayor Ihor Terekhov said.

The port city of Odesa came under missile attack 14 out of 31 days in October, including from ballistic missile strikes aimed at port infrastructure, Ukrainian officials said. More than a dozen people were killed in the strikes.

At least six people were killed and 16 more injured on Tuesday morning in a missile strike in Zaporizhzhia, in southern Ukraine, according to local officials.

And in Kherson, Russian soldiers are hunting Ukrainian civilians with small piloted drones laden with explosives, according to local officials. Many strikes were captured in video footage and broadcast on Russian social media channels.

Two years after the Russians were driven out of Kherson, located on the western bank of the Dnipro River, 25 people were killed and 146 others wounded in Russian attacks in October, four times as many as in the previous month, Roman Mrochko, the head of the Kherson military administration, wrote in a statement.

As the civilian toll grows, the Russian military is intensifying its offensive against Kyiv’s troops in eastern Ukraine.

Ukraine’s top military commander, Gen. Oleksandr Syrsky, warned on Saturday that his forces were confronting “one of the most powerful Russian offensives” since the Kremlin ordered its full-scale invasion.

After driving Ukrainians out of Vuhledar, a former mining town that underpinned Ukraine’s defense of its southern Donbas region for years before falling at the end of September, Russian forces have been advancing at their fastest rate in years.

Just as Russia is using its advantage in manpower and equipment to gain ground in the east, it is hoping to overwhelm Ukrainian air defenses by the sheer scale of its assaults. Private Yurii said that instead of combating Russian attacks one drone or missile at a time, Ukraine ultimately must find a way to take out the Russian weapons at their source.

He recalled how his team once worked 16-hour shifts for 42 days without a break.

“They wear us down this way,” he said. “When people don’t get enough sleep, after a while, their efficiency drops.”

Some missile and drone attacks have had specific targets, like power plants and substations. But Private Yurii said it was not always clear now what the drones were aiming at. Regardless, the Russians are forcing Ukrainian air-defense teams to stay on constant alert, a key part of their strategy, he said.

“They just fly around in circles, sometimes without even a warhead, just a dummy drone, to exhaust resources,” he said.

The constant attacks cannot help but take a toll on both soldiers and civilians.

“Only a fool wouldn’t be afraid,” Private Yurii said. But over three long years of war, he said, Ukrainians have learned not to be paralyzed by fear.

“Russians still can’t come to terms with the fact that we’re still standing,” he said.

Liubov Sholudko contributed reporting. Nataliia Novosolova contributed research.

By dismissing his defense minister, Prime Minister Benjamin Netanyahu of Israel has consolidated his hold over his coalition by removing his main internal critic, making it easier for him to set wartime policy in the short term.

But the move also comes with long-term risks. By firing a popular rival who had opposed some of his most divisive policies, Mr. Netanyahu has fueled criticism that he thinks his personal survival is more important than the national interest.

The departing minister, Yoav Gallant, had broken with Mr. Netanyahu by pressing for a cease-fire with Hamas, saying it was the only way to free dozens of Israeli hostages held by the group in Gaza. On the domestic front, Mr. Gallant had pushed to scrap an exemption from military service for ultra-Orthodox Jews, a measure that risked collapsing Mr. Netanyahu’s government because it angered its ultra-Orthodox members.

“Netanyahu saw Gallant as the opposition within his own coalition,” said Nadav Shtrauchler, a political analyst and former strategist for Mr. Netanyahu. “Now, it will be easier for him to go in his own direction, not just politically, but militarily and strategically.”

Mr. Netanyahu swiftly denied that he would use Mr. Gallant’s departure to fire other senior members of the security establishment. Still, commentators speculated that after replacing Mr. Gallant with Israel Katz, who is expected to be a more pliant defense minister, Mr. Netanyahu would find it easier to remove the military chief of staff, Herzi Halevi.

On a similar note, the re-election of Donald J. Trump on Wednesday may temper any backlash in Washington over Mr. Gallant’s dismissal. The Biden administration saw Mr. Gallant as a trusted partner, especially as its relations with Mr. Netanyahu soured, but the election result has further reduced its influence over the prime minister’s thinking.

“Maybe the Biden administration people didn’t like his firing, but by Jan. 20, you will have Trump,” said Itamar Rabinovich, a former Israeli ambassador to Washington. Mr. Netanyahu is “not very much concerned” by the Biden administration’s frustrations, Mr. Rabinovich said.

Still, the move still comes with potential costs for Mr. Netanyahu.

In firing the defense minister, Mr. Netanyahu has drawn accusations that he is prioritizing personal goals over national ones to appease far-right and ultra-Orthodox members of his coalition. And the move suggested that he will press ahead with policies that are either deeply unpopular, in the case of the exemption for the ultra-Orthodox, or at least polarizing — like his refusal to compromise in the cease-fire negotiations with Hamas.

Neither move will immediately bring down his government, but they could both damage him in a future election.

As Israel fights the longest war in its history, some Israelis are completing their third tours of reserve duty in Gaza or Lebanon. That has raised questions among soldiers about why they should shoulder the burden on the battlefield while Mr. Netanyahu allows ultra-Orthodox Jews to avoid military service.

The resentment over that imbalance could rise over time, even within Mr. Netanyahu’s base, much of which is conservative and religious but still serves in the military.

In a sign of widespread discontent over Mr. Netanyahu’s decision to fire Mr. Gallant, tens of thousands of protesters spilled into the streets in Israel on Tuesday night, blocking a major highway, while newspaper columnists wrote strong condemnations in the Wednesday papers.

“Netanyahu’s sacred principle, his only principle: clinging to power at any cost,” Nadav Eyal wrote in a column for Yediot Ahronot, a centrist newspaper.

“If you, the brave reservist who served 230 days this year, whose children do not sleep at night, whose businesses have suffered, whose relationships with your spouses have suffered — if you have to pay a price so that Netanyahu can close a deal with the ultra-Orthodox, you will pay,” Mr. Eyal added.

The anger has been compounded by recent allegations that Mr. Netanyahu’s office illegally obtained secret documents from the military and leaked them to foreign news outlets in order to torpedo a deal to pause the war in Gaza and free the hostages held there. Mr. Netanyahu has denied the claims and one person in his office has been arrested.

By firing Mr. Gallant, Mr. Netanyahu is “evidently calculating that it will only be a short-term firestorm and he will then be left in a better position,” said Michael Koplow, an analyst at the Israel Policy Forum, a New York-based research group.

But his support for the policies opposed by Mr. Gallant “will cost him in the long term,” Mr. Koplow said. “So even if his short term calculation is correct, he may turn out to be penny-wise and pound-foolish.”

On issues that matter most to Israel’s critics, like the conduct of Israel’s wars in Gaza and Lebanon, Mr. Gallant was fairly aligned with Mr. Netanyahu.

Prosecutors at the International Criminal Court sought arrest warrants for both men in relation to the Gaza offensive. It was Mr. Gallant who played a bigger day-to-day role in managing a campaign against Hamas that has killed tens of thousands of Gazans and damaged most of the enclave’s buildings. Mr. Gallant was also one of the first ministers to push for the killing of Hassan Nasrallah, the Hezbollah leader, months before the government signed off on his assassination.

But within Israel, Mr. Gallant was seen as a thorn in Mr. Netanyahu’s side.

His public disputes with the prime minister began several months before the war, when in March 2023 he spoke out against Mr. Netanyahu’s efforts to overhaul the judicial system. Mr. Netanyahu fired him days later, only to rescind his dismissal after mass protests swept the country.

During the war, Mr. Gallant had spoken publicly about Palestinian governance in postwar Gaza, an idea that Mr. Netanyahu had avoided discussing in detail for fear of angering far-right allies who seek to settle Jewish civilians in the territory.

And Mr. Gallant had developed a strong and independent relationship with the Biden administration, irking Mr. Netanyahu, whose relationship with President Biden has become fractious even as the president continues to arm and fund Israel’s military.

Johnatan Reiss and Gabby Sobelman contributed reporting.

The Kremlin tried and failed for four years to turn President Donald J. Trump’s friendly rhetoric into friendly policy.

Now it has a second chance.

In the run-up to Tuesday’s U.S. election, Russian officials said they cared little about the outcome. American policy toward Russia had only hardened during Mr. Trump’s four years in office, they argued, citing sanctions and his delivery of weapons to Ukraine.

But after Mr. Trump’s victory, the mood began to shift. Some people close to the Kremlin sought to pave the way for rapprochement with Washington despite what many Russians see as an American proxy war against them in Ukraine.

“Trump and his team have a reputation of being very pragmatic,” Kirill Dmitriev, the head of Russia’s sovereign wealth fund and a close ally of President Vladimir V. Putin, said in a phone interview on Wednesday. Mr. Trump’s return to the White House would be an opportunity, he added, to “look at things in a more problem-solving manner than was done by previous administrations.”

Mr. Dmitriev declined to comment on whether he had sent private messages this week to anyone on the Trump team. But he issued a public statement signaling that the Kremlin saw a second Trump presidency as a welcome change, and a new opening to form a bond with Mr. Trump — who has often praised Mr. Putin’s authoritarian leadership and avoided condemning Russia’s invasion of Ukraine.

After the “lies, incompetence and malice of the Biden administration,” Mr. Dmitriev said, there were now “new opportunities for resetting relations between Russia and the United States.”

It was a notable invitation from Mr. Dmitriev, whose role as an informal emissary for Mr. Putin was documented in the American special counsel’s investigation into Russian interference in the 2016 election. At the time, Robert S. Mueller III found, Mr. Dmitriev was already seeking to connect with Mr. Trump’s inner circle the morning after his win over Hillary Clinton.

The Kremlin’s top priority this time around appears to be cutting a deal on its terms in Ukraine. Mr. Trump has said repeatedly that he could end the war in a day, without saying how, and a settlement outlined by Vice President-elect JD Vance echoes what people close to the Kremlin say Mr. Putin wants: allowing Russia to keep the territory it has captured and guaranteeing that Ukraine will not join NATO.

Vladimir Pozner, a longtime Russian and Soviet state television journalist, said in an interview from Moscow that none of his friends and acquaintances had wanted Vice President Kamala Harris to win. Mr. Trump, he said, was seen as someone who could end the war, “probably in Russia’s favor.”

“There is this general feeling that Trump would be better for Russia,” he said. “Maybe we’ll be finished with this war, and maybe the relationship will improve.”

Publicly, the Kremlin strove to strike a muted tone on Wednesday, a contrast to the celebrations of Mr. Trump’s 2016 victory — when there were champagne corks popping in Parliament — that proved premature. While Mr. Trump spoke favorably of Mr. Putin throughout his presidency, American sanctions against Russia increased and his administration was the first to send antitank weapons to Ukraine.

“If someone can change something, then this should be welcomed,” Dmitri S. Peskov, the Kremlin’s spokesman, said, referring to Mr. Trump’s promise to stop the war. “If these are words during the election campaign — we have seen this before.” The Kremlin, noticeably, did not offer congratulations to Mr. Trump, though that may still be in the offing.

Ukraine, of course, would have to agree to any deal that Mr. Trump might try to cut with Mr. Putin, although the United States has leverage as Ukraine’s most important provider of arms. For now, President Volodymyr Zelensky of Ukraine has vowed to keep fighting and says that he will not cede territory; on Wednesday, he became one of the first world leaders to congratulate Mr. Trump by telephone on a “historic” win.

But in Moscow, some are already gaming out scenarios for how Mr. Trump could bring the war to a favorable end. Konstantin Remchukov, a Moscow newspaper editor close to the Kremlin, said the first step would be pushing Ukrainian troops out of Russia’s Kursk region, where they hold a sliver of territory.

After that, he said, Mr. Putin will be ready for talks, conditioned on Russia’s being able to keep the territory it has captured. Mr. Trump might send cabinet designees to make his position clear, even before the inauguration, Mr. Remchukov added. (Any negotiations involving Trump officials before he takes office could be illegal under the 1799 Logan Act.)

“They might say, ‘Let’s have a cease-fire for Christmas,’” Mr. Remchukov said. “And he’s not even president yet, but he’s already racking up the points, because there’s peace everywhere, since he’s the president of peace. That’s how I think it will be.”

During President Biden’s term, the Kremlin still built or kept bridges with people in Mr. Trump’s orbit. Tucker Carlson, the former Fox News host who is part of Mr. Trump’s inner circle, flew to Moscow in February to interview Mr. Putin, becoming the first American media personality to sit down with the Russian leader since his full-scale invasion of Ukraine.

Observers in Moscow point to the connections between Mr. Trump and Mr. Putin, who both espouse conservative, “traditional” values and nurture images as tough, decisive leaders.

“Putin and Trump understand each other to a much greater degree than, say, Putin and Biden,” Mr. Pozner, the television journalist, said. “That’s very much a feeling that a lot of people here have.”

Russian institutions have decriminalized domestic violence, banned the “global L.G.B.T.Q. movement” as extremist and sought to curb abortions — actions that echo policies pursued by Republicans in the United States.

Critics also point to what they call Mr. Trump’s shameful deference to Mr. Putin at a 2018 summit in Helsinki, when he accepted Mr. Putin’s word that he had not interfered in the 2016 election over the assessment of U.S. intelligence agencies.

Yet even some of Mr. Putin’s fiercest opponents said they saw reasons for hope in Mr. Trump’s victory. Ilya Yashin, a prominent anti-Kremlin politician freed in a prisoner exchange with the West in August, said in an interview that “Trump’s first presidency was not so easy for Putin.” Surrendering Ukraine, he said, “would look like an extremely weak decision, and I think Trump understands this very well.”

He added that Mr. Trump’s victory could drive home to Russians that America is a real democracy, not the oligarchy controlled by a liberal “deep state” depicted on their television sets.

“We should be absolutely calm,” Mr. Yashin said. “This is how democracy works.”

During the first administration of Donald J. Trump, if any world leader could claim to have had the now president-elect’s number, it was Shinzo Abe, Japan’s prime minister at the time.

Mr. Abe, who was assassinated by a gunman in 2022, was considered a Trump whisperer par excellence, tapping into the president’s love of golf, hamburgers and adulation in a way that helped shelter Japan from Mr. Trump’s punishing instincts.

Mr. Abe was the first foreign leader to visit Mr. Trump after he was elected in 2016, bringing a gift of golf clubs to Trump Tower in Manhattan. After the president’s inauguration, Mr. Abe quickly assumed the role of elder statesman guiding the new man on the world stage. On multiple phone calls, he was a reliable friendly ear. The first time Mr. Abe visited the new president at his plush resort residence, Mar-a-Lago, just weeks after Mr. Trump took office, the pair played golf together and dined with their wives.

When Mr. Trump came to Japan on a state visit, Mr. Abe piled on the pomp and circumstance, naming a trophy after the president to award at a sumo wrestling tournament and granting the American president the honor of being the first international leader to meet the newly enthroned emperor.

Mr. Abe “moved quickly enough, he got the tone right, he knew how to talk to” President Trump, said Tobias Harris, founder and principal of Japan Foresight, a risk consultancy in Washington. “It’s hard to think of a leader who did quite as well.”

Now, as Japan and the rest of the world brace for the next Trump administration, the question of how to manage the most mercurial of American presidents has officials frantically reviewing their playbooks from those first four years.

Shigeru Ishiba, Japan’s prime minister, picked up the phone Thursday morning in Tokyo to call the president-elect, congratulating him on his “campaign to make America great again” and for “receiving approval from many people.” Mr. Ishiba told reporters that he planned to meet with Mr. Trump as soon as possible.

Mr. Abe’s strategy offers potential guidance on forestalling some of the most aggressive impulses of Mr. Trump’s “America First” agenda. When running for president the first time, Mr. Trump warned that he would impose border taxes on Japanese automakers and assailed Japan for not paying enough for its own defense.

In part because Mr. Abe shared some political values with Mr. Trump — they were both defense hawks and nationalists — and because Mr. Abe knew how to flatter the American president, the Japanese prime minister was able to beat back most of Mr. Trump’s initial threats. Mr. Abe “avoided unnecessary conflicts between the Trump administration and the Japanese government,” said Jiro Yamaguchi, a political scientist at Hosei University in Tokyo.

When Mr. Trump first took office, Mr. Abe had the advantage of having been prime minister for nearly five years already, and Mr. Trump deferred to his more experienced counterpart. When North Korea test-fired a ballistic missile during one of their dinners at Mar-a-Lago in February 2017, Mr. Trump ceded to Mr. Abe during a news conference that followed. Afterward, the president said that he wanted “everybody to understand and fully know that the United States of America stands behind Japan, its great ally, 100 percent.”

As time wore on, observers questioned whether Japan was getting as much as it should from Mr. Abe’s efforts. He had even endured the ignominy of tumbling into a golf bunker while hitting the links with Mr. Trump near Tokyo, yet Mr. Trump left Japan off a list of countries temporarily exempted from tariffs on steel and aluminum imports and failed to keep Mr. Abe in the loop when the U.S. leader decided to accept an invitation to meet Kim Jong-un, North Korea’s leader, for nuclear negotiations.

“The returns were diminishing,” Mr. Harris said.

In the time since Mr. Trump left office in 2020, Japan has taken considerable steps to bolster its defense position. That had been a point of tension with Japan in his first term, when Mr. Trump accused Japan, where about 50,000 U.S. troops are based, of being a free rider on the United States military.

Japan has committed to raising its defense budget to more than 2 percent of its gross domestic product and has rapidly purchased Tomahawk missiles and other missile defense systems from American defense companies.

Because of these actions, said Ichiro Fujisaki, a former Japanese ambassador to Washington, Tokyo is now in a position to tell Mr. Trump “we’ve been doing all that, which he didn’t even have with Abe.”

Yet after a recent general election in which the governing party lost its majority, the government may be in a less stable position to deliver on its defense-spending commitments or to haggle with Mr. Trump over steel tariffs, which President Biden partly lifted in 2022.

Mr. Ishiba, elected just weeks ago, is fighting for his domestic survival and is considered less talented at smoothing ruffled political feathers. Given Mr. Trump’s appetite for entertainment, a “bro” style of communication and open admiration, it is not clear if Mr. Ishiba can manage a second Trump administration as well as Mr. Abe did.

“Ishiba is a serious person who speaks like a university professor,” said Narushige Michishita, a professor of international relations at the National Graduate Institute for Policy Studies in Tokyo. “Ishiba can golf, but I doubt that he can build a similar personal relationship with Trump.”

Hisako Ueno contributed reporting from Tokyo.