The police in the Philippines arrested Rodrigo Duterte, the country’s former president, on Tuesday on charges of crimes against humanity under a warrant by the International Criminal Court.

While in office, Mr. Duterte encouraged the police to hunt down and kill people whom they suspected of being involved in the illegal drug trade. Human rights groups say that more than 30,000 people were killed in extrajudicial executions.

The arrest represents a significant victory for the I.C.C., an independent judicial body that investigates and tries people accused of genocide, war crimes, crimes against humanity and the crime of aggression from its base in The Hague.

But Mr. Duterte’s arrest also shows the limits of the court’s power: Although its jurisdiction is sweeping, the court cannot carry out arrests on its own. It relies on the cooperation of national governments to execute its warrants, which leaves it at the mercy of domestic politics.

Arresting Mr. Duterte was possible because he was out of office and politically weakened. By contrast, there appears to be essentially no chance that the I.C.C. arrest warrants for President Vladimir V. Putin of Russia or Prime Minister Benjamin Netanyahu of Israel will be carried out anytime soon.

On paper, the I.C.C. has the ability to order the arrest of anyone who is alleged to have committed a crime listed in the Rome Statute, the 1998 treaty that established the court, and who is either a national of an I.C.C. member state or committed the crime on the territory of one. Even sitting leaders are not immune from prosecution — a notable departure from the usual rules of international law. (Mr. Duterte withdrew the Philippines from the court during his presidency, but a panel of I.C.C. judges found that the court still has jurisdiction because his alleged crimes took place before that withdrawal.)

But in practice, the court has little to no ability to go after sitting leaders, or people under their protection.

Mr. Duterte may have expected to be protected by his successor, Ferdinand Marcos Jr., who initially pledged to shield him from international inquiry. But then Mr. Duterte’s family — especially his daughter Sara Duterte, who is vice president — had a dramatic falling out with Mr. Marcos (Ms. Duterte said she wanted to cut Mr. Marcos’s head off and threatened to dig up the body of his dead father and throw it into the ocean).

As the relationship deteriorated, so did Mr. Duterte’s protection from the international court. Mr. Marcos began allowing I.C.C. investigators into the country.

Experts say this narrow set of circumstances created a way for the I.C.C. to act.

“We can’t expect a new institution to substantially counteract the operation of power, or real entrenched interests of powerful actors,” said Kate Cronin-Furman, a professor at University College London who studies accountability for mass atrocities. “But what we do hope that they’ll do is adjust things along the margins, and over time shift things toward a more just world.”

“In the absence of the I.C.C., Duterte probably happily lives out his life, responsible for the massacre of thousands,” she added.

Most of the outstanding I.C.C. arrest warrants have not gone the same way. Many of the court’s most high-profile warrants have not resulted in arrests, including the cases of Mr. Putin and Mr. Netanyahu.

For political leaders who are targeted by the I.C.C., the power of their offices can often be enough to protect them from arrest. But with Mr. Duterte as a notable example, that dynamic creates an additional incentive for them to stay in office for as long as possible.

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The U.S. agreed to resume military aid to Ukraine

The U.S. said it would immediately lift the pause on intelligence sharing and resume military assistance to Ukraine after hours of meetings in Saudi Arabia. Kyiv said it would support the Trump administration’s proposal for a 30-day cease-fire with Russia.

In a joint statement, the U.S. and Ukraine acknowledged that the terms of any cease-fire would be subject to Russia’s approval. They also agreed to conclude a deal to develop Ukraine’s critical mineral resources “as soon as possible.”

The U.S. and Ukraine also discussed humanitarian relief efforts that could occur once a cease-fire was in place, and the exchange of prisoners of war. There was no immediate comment from Russia, which did not have officials at the talks.

President Trump told reporters that he might speak this week with Russia’s leader, Vladimir Putin, and that he hoped a lasting cease-fire would be negotiated in the coming days.

What’s next: After the meetings, Secretary of State Marco Rubio said that the agreement now put the pressure on Russia to end the war. “The ball is now in their court,” he said.

Defense: President Emmanuel Macron of France gathered the chiefs of staff of more than 30 armed forces yesterday to review the formation of a multinational peacekeeping force to monitor any cease-fire in Ukraine.

Urban warfare: Russian officials said Ukraine attacked Moscow before dawn yesterday, with at least three people killed and 18 others injured in the broader Moscow region.

Trump threatened Canada, then walked it back

It was a day for confusion and threats. After spending the earlier half of yesterday threatening Canada with doubling tariffs on steel and aluminum, President Trump pulled back after Ontario dropped the surcharges it had imposed on electricity it exports to the U.S. The planned 25 percent tariffs on Canadian steel and aluminum were still set to go into effect, a White House spokesman said.

Markets whipsawed, as investors tried to puzzle through Trump’s commitment to tariffs. Stocks dropped in early trading before recovering late in the day.

Context: Trump’s latest tariff salvo was in response to Ontario’s plan to add a 25 percent surcharge to the electricity it exports to Michigan, Minnesota and New York.

The 51st state: Trump repeated his desire to annex Canada. He said that “the only thing that makes sense” was for the U.S.’s northern neighbor to become its 51st state. He laid out in the clearest terms yet how he envisions that happening, saying it would “make all Tariffs, and everything else, totally disappear.”

More on Trump

Duterte, the former Philippine president, was arrested

Philippine authorities yesterday arrested former President Rodrigo Duterte at Manila’s main airport as he returned from a trip to Hong Kong, acting on a warrant issued by the International Criminal Court. The I.C.C. has accused Duterte of crimes against humanity for his past antidrug campaigns, which rights groups say killed roughly 30,000 people, many of whom were not involved with the drug trade.

What’s next: Duterte was flown on a plane bound for The Hague, where the court is based. The case will test the legal reach of the I.C.C., which draws its jurisdiction from the Rome Statute, a treaty currently signed by 125 countries. Here’s what else to know.

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Kigali, Rwanda’s capital, has presented itself as an orderly city with ample attractions and premier sporting events. But that image, and Rwanda’s position as a security partner in Africa, are increasingly at odds with the war that has killed thousands in the Democratic Republic of Congo, where Rwanda backs M23, a rebel group. Tourism brought Rwanda $620 million in 2023, but sanctions are piling up.

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As the producers of the hit West End musical “Operation Mincemeat” prepared to go to New York, they asked a question: “Are we too British for Broadway?” The answer, they hoped, would come from Americans before opening night, later this month.

The show, a comedy about a World War II spy operation, went across the pond armed with an Olivier Award for best new musical and a request for preview audiences to email feedback to the director. Tweaks were made for Broadway palates.

“American audiences are quick and sophisticated, but they also love the dumb stuff, like the slapstick,” the director said. “Almost more than the British.”

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Portugal’s Parliament ousted the center-right government in a vote of no confidence on Tuesday, forcing the third national election in little more than three years.

The year-old government of Luís Montenegro was precarious from the start, controlling far less than a majority in the Parliament, and it has recently been tarnished by a controversy over his business dealings.

President Marcelo Rebelo de Sousa said that the election to form a new government could take place in May.

Mr. Montenegro’s Democratic Alliance, a coalition of conservative parties, finished first in the March 2024 election, but with only 29 percent of the vote, barely ahead of the Socialist Party. The hard-right Chega party, just a few years old, secured 18 percent of the vote to finish third.

With no party able to assemble a majority coalition, Mr. Montenegro formed a minority government.

Last month, Portuguese newspapers reported that the prime minister’s family owned a real estate and consulting company, and that among its clients was a casino operator whose concession contracts depended on government decisions.

Mr. Montenegro said he had transferred his ownership stake to his wife in 2022, and had not profited directly from it since taking power. But opposition parties said the revelations raised concerns about potential serious conflicts of interest and demanded investigations, prompting the vote of confidence that the government lost on Tuesday.

The Socialist Party, which governed from 2015 to 2024, and Chega stand to benefit from the damage to the mainstream conservatives, but both of those leading opposition parties have been accused of ethical failings of their own.

A Chega lawmaker was charged in January with stealing luggage at Lisbon’s airport, and in February a senior party member was accused of paying a minor for sex, a blow to the party’s law-and-order image. Both were expelled from the party.

In January 2022, the Socialist Party won an absolute majority of seats in the Parliament, enabling it to govern without coalition partners. But in November 2023, Prime Minister António Costa’s name was linked to an investigation of suspected ties between government officials and lithium and hydrogen exploration businesses, leading to his resignation and snap elections.

Mr. Costa was not charged, and has since become president of the European Council, the group of European Union heads of government, but a former chief of staff of his does face charges in the case.

Russian forces have begun an assault on the town of Sudzha, the main population center in the Kursk region that was captured by Ukraine last year, a top Russian commander said on Tuesday. The advance puts Moscow on the verge of ending the first invasion of Russian territory since World War II.

“We have surrounded Sudzha from almost all sides,” Lt. Gen. Apti Alaudinov, a top Russian commander in Kursk, told Russian state television on Tuesday.

Russia’s recent rapid moves around Sudzha have erased a large share of the territorial gains made by Ukrainian forces during their surprise attack into western Russia in August. Kyiv had held on to territory in Kursk despite heavy casualties in the hope of using it as a bargaining chip in peace talks.

Amid Ukraine’s advance, 2,000 to 3,000 Russian civilians had taken refuge in Sudzha by February, according to Russian activists and Ukrainian officials. Their fate is unclear.

By Tuesday afternoon, Ukrainian forces had retreated from the eastern side of town across the Psel River, Ian Matveev, an antiwar Russian military analyst, wrote on his Telegram channel. He cited videos posted on social media of Russian soldiers in the town. Deepstate, a Ukrainian group with ties to the Ukrainian Army that maps the battlefield, made a similar assessment on Tuesday.

Mr. Matveev wrote that it remained unclear whether Ukrainian soldiers would try to mount a defense in the western side of Sudzha or continue retreating toward the Ukrainian border, a few miles west.

On Tuesday morning, Russian state media published a photo of a Russian soldier standing at the eastern entrance of Sudzha, a county seat in Kursk that became the main logistical and administrative hub for Ukraine’s occupation forces. The New York Times verified the location of the photo.

On Tuesday morning, the Russian Defense Ministry claimed that its forces had liberated 12 Kursk villages and 38 square miles in the past 24 hours, and analysts speculated that Ukraine’s presence in Russia could end within days.

The Kursk offensive is part of a wider effort by Russia and Ukraine to improve their positions ahead of peace talks mediated by the Trump administration, the Russian military analyst Valery Shiryaev wrote on his Telegram channel on Monday.

As Moscow’s forces pressed forward in Kursk, Ukraine on Tuesday launched the largest drone attack on Russian cities since the start of the war, a move that analysts interpreted as an attempt by Kyiv to show Washington that it can still take the fight to Russia.

“Everyone is putting on the table the cards that they have,” Mr. Shiryaev wrote.

The rapid crumbling of Ukrainian defenses around Sudzha signals an end to Kyiv’s determined attempts to hold on to captured Russian territory, which at its peak measured more than 500 square miles. To keep the territory, Ukraine had transferred to Kursk some of its best reserves, a strategy that critics said weakened Kyiv’s ability to defend itself against Russian attacks in eastern Ukraine.

Ukrainian officials have repeatedly defended their strategy in Kursk in political terms, saying that territory could be traded for land that Russia has occupied in Ukraine — a possibility that was vanishing rapidly by Tuesday.

Russian soldiers and commanders fighting in Kursk previously said in interviews that they had expected the battle for Sudzha to last months and become one of the bloodiest standoffs of the three-year war, which began with Russia’s full-scale invasion of Ukraine in 2022. It had taken Moscow’s soldiers half a year to grind through Ukrainian defenses and reach the outskirts of the town earlier this year — at the cost of thousands of casualties by most estimates.

Ukraine’s glacial retreat suddenly turned into a rout in the past week.

Russian military analysts said the breakthrough in Kursk began on Saturday with a surprise attack on the Ukrainian rear. Russian officials said 800 fighters walked about 10 miles through a disused gas pipeline measuring 4.5 feet wide to emerge inside Ukrainian-held territory and engage the enemy in Sudzha’s northern suburbs.

Russian propagandists and officials described the operation as a heroic feat, while Ukrainian sources called it a death sentence, with some Russian attackers allegedly suffocating from residual methane in the pipeline. That claim could not be independently verified.

“I respect them for this creativity,” said a Ukrainian military official who spoke on the condition of anonymity because he was not authorized to speak publicly. He added that the Russian attackers had suffered heavy casualties and that only 90 had made it from the pipeline into Sudzha.

While the effectiveness of the pipeline operation is debatable, it coincided with breaches of Ukrainian defenses by Russian troops in several parts of Kursk.

Ukraine’s top military commander, Gen. Oleksandr Syrsky, said on Monday night that Kyiv was dispatching reinforcements to Kursk, but rejected Russian claims that a large contingent of Ukrainian soldiers there was at risk of encirclement.

“A decision was made to reinforce our group with the necessary forces and resources,” he said. He added that Ukrainian forces were retreating to more favorable defensive positions, which some commentators interpreted as a possible prelude to a complete withdrawal from Kursk.

Russian military analysts said a withdrawal was likely to merely shift the battle to Ukrainian territory, pitting Kyiv’s forces in the Sumy region that borders Kursk against tens of thousands Russian soldiers and their North Korean allies.

Sanjana Varghese, Marc Santora and Michael Schwirtz contributed reporting.

In “Romeo and Juliet,” the star-crossed heroine asks: “What’s in a name? That which we call a rose, by any other name would smell as sweet.”

People far less infatuated have wrestled with this concept for hundreds of years: How much should a name signify, and does it actually affect what or who a person is?

Meghan, Duchess of Sussex, would like a word.

In her new Netflix series, “With Love, Meghan,” the duchess, who is married to Prince Harry, told the actress Mindy Kaling that her last name was Sussex, correcting Ms. Kaling, who had referred to her by a more familiar name: Meghan Markle.

“It’s so funny, too, that you keep saying Meghan Markle,” Meghan said in the second episode of the series, which premiered last week. “You know I’m Sussex now.”

Meghan cited the importance of sharing a last name with her children.

“I didn’t know how meaningful it would be to me but it just means so much to go, ‘This is our family name, our little family name,’” she said.

Ms. Kaling, who initially seemed surprised, replied, “Well, now I know and I love it.”

It’s understandable that Meghan, whose representatives did not immediately respond to a request for comment, insists on usage of what she feels is the correct form of her name. But as with most Meghan-related news, the clip quickly made waves online as people took to social media to criticize her. Some commenters thought she was being pretentious, and others called her out for seemingly having confused her royal house with the family’s surname.

There was similar confusion over how to refer to the duchess after she and Prince Harry announced in 2020 that they would “step back” from their official royal duties and move to the United States.

Two days after the new series premiered, during an appearance on “The View,” Ms. Kaling said that she had “a great time” on the show, despite critics who felt like Meghan had behaved in a passive aggressive way toward her.

There is still some debate, however, on Meghan’s last name.

“She’s either totally oblivious to what her actual name is, she doesn’t understand it or she’s lying,” Hilary Fordwich, a royal family expert, said in a phone interview.

Ms. Fordwich explained that while it isn’t new for a member of the royal family to choose to go by their birth titles (Prince Harry went by Harry Wales during his time in the British army), that doesn’t make it their family name.

“By established protocol, he can use Harry Sussex, which I’ve never heard him use,” she said. “She could choose to use Meghan Sussex, but it’s not their surname. This was the issue in that Netflix clip.”

However, Rachel Bowie, the royals editor for the lifestyle website PureWow, didn’t think what Meghan said was problematic at all, adding that it’s “totally within royal protocol” and she didn’t see it as a formal change.

“Even though Archie and Lili were christened ‘Mountbatten-Windsor,’ Harry and Meghan are borrowing Sussex from their title to make up their last name,” Ms. Bowie, who was previously the co-host of the “Royally Obsessed” podcast, said in a phone interview.

“I never read it as Meghan formally changing her name, but more as this symbolic thing for herself, that she feels this connection, between the four of them, that they move through the world under the name Sussex,” she added.

Opinions have continued to roll in, even from relatives of Harry and Meghan.

Lord Ivar Mountbatten, a cousin of King Charles III and a reality television star on “The Traitors,” claimed she had been wrong about her own surname in an interview he gave to Town & Country, saying the family’s surname is Mountbatten-Windsor.

“Her children are called Archie and Lilibet Mountbatten-Windsor; they’re not called Archie and Lilibet Sussex, because Sussex is a title,” he said.

And Thomas Markle, Meghan’s estranged father, complained to the Daily Mail on Saturday about her decision to no longer use the name Markle, despite the fact that it’s still fairly common for a woman to adopt her husband’s name after getting married.

Understanding exactly how to refer to the royal family requires a look at its evolving history regarding names. According to the official website for the British Royal Family, “Members of the Royal Family can be known both by the name of the Royal house, and by a surname, which are not always the same.” It’s uncommon for core members of the Royal Family to be referred to by a surname at all.

Members of the Royal Family had no surname at all before 1917 and were referred to only by the name of the house or dynasty they belonged to. That year, King George V adopted Windsor as the name of the house and surname of his family.

In 1960, Queen Elizabeth II and Prince Philip, Harry’s grandparents, altered the name to distinguish their direct descendants from the rest of the extended Royal Family, making their new surname Mountbatten-Windsor, which included Philip’s family name.

“For the most part, members of the Royal Family who are entitled to the style and dignity of HRH Prince or Princess do not need a surname, but if at any time any of them do need a surname (such as upon marriage), that surname is Mountbatten-Windsor,” the site reads, using the abbreviation for His or Her Royal Highness.

Meghan now lives outside that structure, meaning those rules do not necessarily apply to her. So her choice to be referred to as Meghan Sussex, for now, will simply be added to the list of things she’s either criticized, or embraced, for.