Noncitizens who are not required to register with Selective Service include men who are in the U.S. on student or visitor visas, and men who are part of a diplomatic or trade mission and their families. Almost all other male noncitizens are required to register, including illegal aliens, legal permanent residents, and refugees.
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Dual nationals of the U.S. and another country are required to register, regardless of where they live, because they are U.S. nationals.