“At the moment you can take your pet dog, cat, or indeed ferret, from the UK to the EU and back again without quarantine provided that certain conditions are met, such as having a pet passport and your pet being microchipped.
Pet passports are issued by EU countries and a short list of other countries such as Greenland, Iceland and Switzerland.
The UK could be added to this list, but clearly agreements would be needed to make that happen – it would not be automatic.
For pets from countries that do not issue passports, the EU distinguishes between listed and non-listed countries, depending on whether they have suitably robust surveillance and reporting systems for diseases such as rabies.
Pets from listed countries such as the US, Japan and Russia, need to have paperwork filled out, including health certificates from their vets, if they are to avoid blood tests or quarantine at the border. They will have had to have a test for rabies within three months of the date of travel, for example.” – 7 January 2018 http://www.bbc.com/news/uk-41969672