The below version is specific to Ireland and my experience as at writing of this (May 2019)
Disclaimer: This below is my experience importing my dogs into Ireland, may differ from others experience. Do not see this advise and ensure that you verify facts on the various websites (see links further down) for the latest information. I don’t know the rules regarding cats and other animals, but know that with cats it is very similar to dogs. I wrote this from my experience with dogs and sometimes mention dogs often but read this document as Animals/Pets.