As Mr. Trump walked through a hallway in the Knesset, Israel’s Parliament, a reporter asked if the war was officially over. “Yes, as far as I’m concerned, yes,” he replied.

But the cease-fire deal did not end the war, and there are many lingering questions about what comes next in Gaza, including whether Hamas will agree to Prime Minister Benjamin Netanyahu’s demand that it disarm. Many of those questions were expected to come up at a summit on the cease-fire deal that Mr. Trump and other world leaders were scheduled to attend in Egypt later on Monday. The Egyptian government said that Mr. Netanyahu and Mahmoud Abbas, the president of the Palestinian Authority, would participate in the summit, a rare meeting between the two as the peace process between Israelis and Palestinians has stalled.

In both Israel and Gaza, the cease-fire and the start of the exchange on Monday brought relief after two devastating years of war.

Israeli television showed Einav Zangauker, the mother of Matan Zangauker, 25, speaking on a video call with her son in Gaza for the first time since he was abducted two years ago from his home in Nir Oz, near the Gaza border. “There’s no war, it’s over,” she told him. “You are coming home!”

But for some Gazans, the relief was clouded by sadness over a war that has reduced much of the territory to rubble. Saed Abu Aita, 44, said he hoped the return of the hostages to Israel would open the door to the end of the war. But “there’s nothing to be happy about,” he said. “My two daughters were killed, my home was destroyed, and my health has deteriorated.”

Hamas militants attacked Israel in October 2023, killing about 1,200 people and abducting about 250. In response, Israel invaded Gaza, killing about 67,000 Palestinians, according to Gazan health authorities.

For Israelis, getting the hostages back was by far the most important goal. Scores of hostages were released in two earlier temporary cease-fires, and a small number were rescued alive by Israeli soldiers.

For Palestinians, the cease-fire offered hope of an eventual return to something approaching normal life. Thousands of displaced Gaza residents began to return north to Gaza City after it took effect. Israel has also begun to allow growing amounts of desperately needed food, medicine and other humanitarian aid into the enclave, where a United Nations-backed panel of food experts has said that some areas are officially under famine.

Here’s what else to know:

• The logistics: Under the deal reached last week in talks brokered by U.S., Arab and Turkish mediators, Israeli forces withdrew to a new defensive line inside Gaza by noon on Friday, opening a 72-hour window for the hostages to be released. Read more ›

• The dead: Hamas is required under the cease-fire to turn over the remains of deceased hostages, but it is unclear how quickly that will happen. Israel believes 26 hostages are dead, while the condition of two other captives has not been confirmed. Officials have said that Hamas will have to search for the remains of some hostages, which may take some time.

• Aid deliveries: Hours before the first Israeli hostages were released, the United Nations said that “real progress” was being made in delivering aid to Gaza. Read more ›

• Red Cross: The International Committee of the Red Cross plays a central role in exchanges of hostages in Gaza for Palestinian prisoners and detainees in Israeli jails. The group has come under sharp criticism from both Israelis and Palestinians. Read more ›

• Overnight gathering: Thousands of Israelis crammed into a plaza in central Tel Aviv known as Hostages Square to show support by waving Israeli flags and signs bearing photos of those expected to be freed. Read more ›

“The easing of restrictions on movement and access in multiple areas” had allowed the agency to deliver medical and emergency supplies where they were most needed and support displaced families to prepare for winter, the agency said.

Cooking oil had entered the territory for the first time since March, and tents, flour, frozen meat, fresh fruit and medicines crossed into Gaza throughout the day on Sunday, the agency said. It added that, alongside its partners, it had distributed hundreds of thousands of hot meals and bread bundles in the north and south of the territory.

Humanitarian aid has been one of the most bitter flashpoints of the war. A U.N.-backed panel of food experts said parts of Gaza were officially under famine, and aid groups have accused Israel of imposing onerous rules and restrictions on what items can enter Gaza.

Israel has rejected the panel’s findings. Israeli officials have said they allowed enough food into the territory but claimed, without proof, that much of it was systemically stolen by Hamas.

The U.N. said in its statement on Sunday that it had also secured Israeli approval for more aid to move forward, including water, shelter, medicine and other supplies.

Tom Fletcher, the top U.N. humanitarian official, said last week that almost the entire population of Gaza needed some form of food aid, including 500,000 people who needed treatment to address the effects of famine. The agency had a plan to scale up delivery of aid over the first 60 days of the cease-fire, he said.

The U.N. plans to send hundreds of trucks into the territory each day to provide food, increase efforts to provide shelter, establish more health care sites and dispatch emergency medical teams into Gaza, and help repair basic municipal services like clean water, a functioning sewage system and garbage pickups, Mr. Fletcher said.

The 200-by-300-foot plaza in front of the Tel Aviv Museum of Art became the focal point for demonstrations after a sit-in outside Israel’s military headquarters, a block away, in the weeks after the Hamas-led attack on Israel on Oct. 7, 2023, swelled and needed to be moved.

The plaza has hosted art installations about the hostages’ plight, including a long rectangular table set for 234 people that is surrounded by empty chairs to represent the captives. After some of the hostages came home, parts of the table were set with moldy bread and bottles of dirty water, to correspond to the conditions of captivity they described.

There have also been dance lessons, yoga classes and impromptu singalongs held in honor of the captives, as well as commemorations for missed birthdays.

For relatives of the captives, the plaza became a sort of home, and a place to find company in other people who could understand what they were experiencing. Many set up tents in the plaza, and a steady stream of Israelis and tourists visited to show support.

During an earlier cease-fire in February, families of the captives and their supporters went to the plaza each week to watch live broadcasts of hostage releases.

On Saturday night, it was where hundreds of thousands of people celebrated the announcement of a new cease-fire deal and a possible end to the war.

Ephrat Livni and Roni Caryn Rabin contributed reporting.

“Reducing the role of the I.C.R.C. as just a transport service really misses the bigger picture of what we do,” says the group’s website. “When we transport those affected by armed conflict, it is never just about moving people from one place to another.”

The Red Cross describes itself as a “a neutral, independent organization” that ensures humanitarian protection and help for people affected by armed conflict. Since the war in Gaza began more than two years ago, it has helped to coordinate hostage-prisoner exchanges during two negotiated truces, one in November 2023 and one in January of this year.

The group has come under sharp criticism from both Israelis and Palestinians during the war in Gaza.

The families of some of the hostages taken from Israel have accused the I.C.R.C. of failing to visit the in captivity to check on their health and provide medications, and of not doing enough to work for their release. The organization long said that it did not know where the hostages were being held and that much of its work was carried out behind the scenes.

The group has also been criticized for not visiting Palestinians detained in Israel since the Hamas-led attack on Oct. 7, 2023, that ignited the war in Gaza. The Red Cross said it was denied permission to visit the detainees, but that it had engaged with more than 1,250 people who were released in the exchanges between Israel and Hamas “to gather information on their treatment and conditions of detention.”

The organization has also been criticized for remaining neutral about such a deeply divisive conflict. The Red Cros says its neutrality allows it “to safely transfer hostages and detainees, protecting them during the vulnerable moments” and argues that it “is the trust placed in us by the Israeli authorities and Hamas that enables us to work with all involved to carry out these delicate operations.”

“Neutrality provides us with the ability to try and influence both sides to act in accordance with the laws of war,” Julien Lerisson, head of the group’s delegation in Israel and the occupied territories, wrote in March. “It is the confidential diplomatic work — in which things are said clearly, behind closed doors, without publicly denouncing or supporting either side — that makes humanitarian work possible.”

During the cease-fire in January, Hamas staged ceremonies involving some of the released hostages, drawing widespread international condemnation. Hamas, by then badly battered by the war and falling in many Gazans’ estimation, used the handover of hostages to portray itself as a still-powerful force. For example, Hamas held a “signing ceremony” between one of its members and a representative of the Red Cross before handing over a group of four women being held as hostages.

Mr. Lerisson later explained that he had decided to continue with the releases, despite such “difficult scenes,” so as not to risk the lives of the hostages forced to engage in the ceremonies or those still waiting to be released.

On Friday, the I.C.R.C. president, Mirjana Spoljaric Egger, welcomed the cease-fire agreement between Israel and Hamas, urging both to stick to their commitments to the deal brokered by international mediators.

“Release operations must be carried out safely and with dignity,” she said. “Humanitarian assistance must resume urgently at full capacity and be delivered to people safely wherever they are. The cease-fire must hold. Lives depend on it.”