Generally, the Centers for Disease Control and Prevention regulates the importation of pets into the United States. The Department of Agriculture has additional restrictions on some types of dogs imported to work livestock. The CDC does not require general health certificates, though they are required by some airlines and some states.
Pets taken out of the United States are subject to the same regulations as those entering for the first time. Pet dogs and cats are subject to inspection on arrival for evidence of zoonotic disease (a disease that can be transmitted to humans). If they show evidence of a disease that may be transmissible to humans, they are subject to veterinary medical examination, treatment, and/or quarantine.
Dogs and cats arriving form countries where rabies is reported are required to be immunized against rabies. Dogs and cats arriving from countries considered by the World Health Organization to be rabies free are not subject to the rabies vaccination requirement, provided they have spent 6 months prior to arrival in a rabies free country.
Dogs and cats arriving in the state of Hawaii, the territory of Guam are subject to additional requirements.
It is strongly recommended that all domestic dogs and cats receive periodic rabies vaccinations.
Unless they are included in a specific embargo from the CDC or the U.S. Fish and Wildlife Service, these import of these animals are not regulated at a national level, though some animals may be restricted at a state level (ferrets in California for example).
The U.S. Department of Agriculture, Animal Plant Heath Inspection Service does not have animal health requirements for any of the following, provided they have not been inoculated with any pathogens for scientific purposes: fish, reptiles, lions, tigers, bears, mink, rabbits, sugar gliders, foxes, monkeys, endangered species, guinea pigs, hamsters, gerbils, mice, rats, chinchillas, squirrels, mongoose, chipmunks, ferrets, or other rodents.
After it was discovered in 1975 that small turtles frequently transmitted salmonella to young children, the CDC imposed a limit of 6 turtles under 4 inches of length. U.S. Fish and Wildlife Service regulates the import of reptiles.
Information gathered from the Centers for Disease Control and Prevention, The U.S. Department of Agriculture, and the U.S. Fish and Wildlife Service.