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Under U.S. law, the government is obliged to allow people from other countries to enter the U.S. to escape persecution in their home country. A person who has been persecuted, or reasonably fears persecution based on race, nationality, religion, political opinion, or membership in a particular social group (the last one being a bit of a catch-all) can apply for asylum in the U.S., even if they have entered the U.S. illegally. However, if your trip to the U.S. involved stops in other countries which are just as safe, to which you could have applied for asylum, your request is likely to be denied. Once you arrive in the U.S., you must apply for asylum within one year. If you wait too long, your application is very likely to be denied.
If it is shown that the applicant participated in any of the persecution which they are now fleeing, they cannot apply for asylum. Furthermore, if they can be shown to have recently committed an aggravated felony (a very serious crime, such as murder or rape), they will have their application denied.