U.S. immigration laws place no limit on the number of people who can be awarded asylum in the United States each year. The number of actual asylum grants varies, depending on how many people apply (of their own volition or as a defense to deportation), and how many are successful with their asylum claims.
Don't be confused by the fact that only a limited number of people can be awarded refugee status each year. The U.S. president establishes this annual limit. Although the grounds for receiving asylum status and refugee status are the same (per the refugee definition found in § 101(a)(42) of the Immigration and Nationality Act or I.N.A., or 8 U.S.C. § 1101(a)(42)), the procedural requirements are different, and these two categories are treated differently.
See below for further discussion of the distinction between asylum status and refugee status.
The United States provides refuge to people from other countries who are unable or unwilling to return to those countries because they have either been persecuted in the past or have a "well-founded" fear of future persecution. The persecution may have been by either their government or by forces beyond the government's control. The basis for the persecution must be either the applicant's race, religion, nationality, political opinion, or membership in a particular social group (or a combination thereof).
The number of asylees tends to run about 25,000 per year. To get the latest figures on how many people were awarded asylum as well as refugee status in given years, go to the Department of Homeland Security's website, which includes a page on Immigration Statistics. Look for the latest "Refugees and Asylees" report.
Asylum allows people to apply for protection upon or after their arrival in the United States (though under the Trump Administration, people attempting to apply at the Mexican border have been either placed into detention facilities or forced to wait in Mexico while applying).
The law says that people can apply for asylum whether they arrive legally (with a visa and inspection at a border or other point of entry) or illegally (without inspection). They can either submit an affirmative application within a year of entry or apply as a defense to either expedited removal (upon entry) or other placement in removal proceedings (for example, after apprehension by U.S. immigration authorities within the U.S.).
By contrast, people who are outside the U.S. and seek protection based on past or future persecution must apply for refugee status, usually with the help of the U.N. High Commissioner for Refugees, and through overseas offices of U.S. Citizenship and Immigration Services. There is no mechanism by which to specifically request refugee status in the United States. See a detailed description in How to Prepare and Submit a Refugee Application to the U.S.
In the past, the annual limits on refugees set by the U.S. president tended to range from between 70,000 and 90,000 admissions. The Trump Administration drastically reduced this, setting an annual limit for 2021 of 15,000 refugee admissions. The Biden administration is in the process of raising the number to 125,000 annually starting in 2022. However, because the refugee processing agencies' operations had been drastically scaled back during the Trump Administration, progress could be difficult.
People granted asylum receive the to right to live in the U.S. for as long as they qualify as asylees. They also may apply for a work permit. After a year of approval, asylees can apply for a U.S. green card (lawful permanent residence) through the process known as adjustment of status.