Nearly 100,000 migrants have arrived in New York City since the spring of 2022 as of early August.

The city has struggled to respond as people from all over the world have arrived, sometimes by the hundreds each week. Many have sought shelter with the city, which has a legal obligation to give beds to anyone who asks. Last fall, the city’s homeless shelter population hit a record. It has only grown since then.

Mayor Eric Adams has called it a humanitarian crisis that will cost the city about $12 billion over three years. In the fall, he declared a state of emergency. In recent weeks, city officials have said they are running out of room.

The mayor has repeatedly asked the federal government — even traveling to Washington — for more funds and for expedited work authorizations for migrants, so they can become self-sufficient. He has said that President Biden has “failed” the city by not doing more.

“While New York City will continue to lead, it’s time the state and federal government step up,” Mr. Adams said at a recent news conference.

The vast majority of Venezuelans are staying in other countries in Latin America and the Caribbean. But many Venezuelans have been making the long and dangerous trip to the United States in recent months. About 100 Venezuelans were apprehended annually at the border between 2015 and 2018. More than 150,000 were apprehended between October 2021 and the end of August 2022.

More recently, a large number of migrants have also been coming from countries in Africa.

After crossing the southern border, thousands have made their way to New York with the help of officials in Texas. Gov. Greg Abbott has sent thousands in a campaign to provoke outrage and force the federal government to tighten border security. But Mr. Abbott’s buses account for only a small fraction of the people who have arrived; El Paso, a Democrat-led city, has also sent new arrivals to New York at the migrants’ request, officials there have said, and some people have made their own way.

How is the city responding?

In March, the city announced the creation of a 24-hour center to welcome migrants and a new agency to help coordinate the arrival of asylum seekers, but the city’s response has, at times, been fragmented and reactive as the shelter system has become more strained.

More than 57,000 migrants were staying in city homeless shelters by August, Anne Williams-Isom, the deputy mayor for health and human services, said this week. In total by then, 108,400 people were staying in homeless shelters across the city, officials said.

The city has proposed using a variety of locations as emergency housing for migrants. It has housed people in hotels, emergency tent shelters on Randall’s Island, school gymnasiums and office buildings and is now looking to new places, like the parking lot of a state psychiatric hospital.

Many of the proposals have been met with pushback from residents, and in some cases the city has retreated. At one point, Mr. Adams seriously considered housing migrants on cruise ships. In all, the city had opened 194 shelters for asylum seekers by early August.

As more and more migrants have arrived, Mr. Adams has changed his messaging and his approach to sheltering them.

He has asked a judge to relieve the city of some of its legal obligations under its unique “right to shelter” law. He used an executive order to suspend some of the requirements under the law this spring, in anticipation of an influx of new migrants.

In recent months, the Adams administration has discouraged migrants from coming, distributing fliers at the southern border telling migrants that there is “no guarantee” they will receive shelter or services.

“Housing in NYC is very expensive,” the fliers said. “Please consider another city as you make your decision about where to settle in the U.S.”

The city has also helped migrants leave the city for other counties in New York, angering some officials in other parts of the state.

How much is caring for migrants costing the city?

The city has estimated that it would spend about $5 billion this fiscal year to house and feed migrants. And in a news conference on Wednesday, Mr. Adams said the cost would exceed $12 billion over the next three years, if migrants continued to arrive at the same rate.

Gov. Kathy Hochul has said that she would ask the State Legislature to allocate $1 billion in the next budget to help the city. The state has already given $1 billion, she said, and is paying for a new tent shelter on Randall’s Island.

The city’s spending has sometimes come under scrutiny.

DocGo, a medical services firm that once contracted with the city to provide Covid testing and vaccinations, has moved hundreds of migrants outside the city under a no-bid $432 million contract. The contract called for the group to house migrants and to provide them with food and services like case management, transportation and round-the-clock security. But migrants have said they were lied to, and that representatives of the company gave them documents that falsely claimed they were eligible to work.

What will happen to the migrants next?

Many migrants have said they are pursuing asylum. But it can take three to four years before a final decision is made in asylum cases, which are often complicated and plagued by delays.

Between March and May of this year, nearly 39,000 new immigration court cases were filed in New York City, compared with about 11,000 in Miami-Dade County, Fla., and about 16,000 in Los Angeles County, according to the Transactional Records Access Clearinghouse at Syracuse University.

Applicants can apply for temporary employment authorization 150 days after successfully filing an asylum application, but are not eligible to receive a work document until then. (Mr. Adams has asked for President Biden to provide a path to expedited work authorizations for newly arrived migrants, allowing them to work legally.)

The city employs caseworkers to connect with newly arrived asylum seekers and help them enroll their children in school and find immigration lawyers. And in June, Mr. Adams announced the creation of the Asylum Application Help Center, which will bring together immigration legal service providers and pro bono lawyers and aims to serve thousands of asylum seekers.

But even as migrants have applied for asylum in record numbers, advocates and immigration lawyers say that without additional legal support, many — perhaps the majority — will miss their application deadline and fall into a more perilous category of immigrant: the undocumented.

Already, many new arrivals in New York have found jobs in the underground economy and joined the existing pool of undocumented workers, where they remain vulnerable to deportation and exploitation.

Will the flow of migrants continue?

The Biden administration announced in the fall that up to 24,000 Venezuelans would be accepted into the country through a humanitarian parole program. The program, similar to one established for Ukrainians, would require Venezuelans who apply to have someone in the United States able to support them financially for up to two years.

In the days after the program took effect, the number of Venezuelans who entered the country through the U.S.-Mexico border plunged, and some migrants were stranded in Mexico and other countries.

The Biden administration also announced a new asylum policy in an effort to stem illegal crossings this spring. The policy disqualified most people from applying for asylum if they have crossed into the United States without either securing an appointment at an official port of entry or proving that they had sought legal protection in another country along the way.

The policy was struck down by a federal judge in July, but was upheld on appeal while legal challenges to the policy work their way through the courts.

The number of people crossing the border dipped as a result, but in July appeared to be on the rise again.

Emma G. Fitzsimmons, Juan B. Garcia, Jasmine Sheena and Téa Kvetenadze contributed reporting.