Carry-on pets are not allowed on transatlantic flights. Flying with your dog doesn't have to be a stressful experience if you make plans in advance, ask a lot of questions and thoroughly prepare your dog. Proof of disability may be required. When traveling to other countries, check the local regulations regarding service dogs. Make sure to get a current health certificate and an updated certificate of vaccinations for your dog.
Most destinations require rabies shots.
A few destinations, such as Hawaii and England, either have a 90-day quarantine period or completely prohibit pets from visiting. Write it all down. Arrive at the airport in plenty of time with a well-exercised, pottied, comfortable dog. Put extra food and water in a pack on the back of the dog's crate. If the weather is hot, use a frozen cold pack under the wire grate of the crate.