Asylum
The ability to immigrate to the U.S. on the basis of an asylum request is recognized in the U.S. The U.S. was first populated in significant part by political and religious refugees, and as a result, the people of the U.S. have always been especially sympathetic to asylum requests. A large body of immigration laws and procedures deal with this issue.
According to the Immigration and Naturalization Act (INA): The Attorney General (AG) shall establish a procedure for an alien physically present in the U.S. or at a land border or port of entry, irrespective of such alien's status, to apply for asylum, and the alien may be granted asylum in the discretion of the AG if the AG determines that such alien is a "refugee" within the meaning of the law.
A "refugee" is defined as "any person who is outside any country in which such person last habitually resided, and who is unable or unwilling to avail himself or herself of the protection of that country because of persecution (or a well-founded fear of persecution) on account of race, religion, nationality, membership in a particular social group, or political opinion." From these two statements has grown a very sizable body of law, regulations and procedures as well as case law.