If you Wondering On domestic flights carried by American Airlines, passengers are allowed to bring small pets into the cabin. American Airlines Pet Policy states that each passenger is only allowed to bring one pet carrier on board. Your carrier will be considered the customer's carry-on bag or can carry up to 2 cats or dogs of the same species. To fit under the seat in front of you, it needs to be compact. For pets to fly, Southwest does not require a health certificate or any other papers from the client's veterinarian. Pets are sadly not permitted to go on overseas flights.