Post-Brexit, EU PETS passports that were issued in the UK are invalid for travel to EU countries. However, they are still valid for entry into the UK from the EU.
PETS passports issued in EU countries are still valid for travel back forth between the UK and the EU.
If you are in the UK, unless you have a PETS passport that was issued in an EU country, you will need an Animal Health Certificate (AHC) to take your dog or cat to an EU country. For each trip to the EU, you will need a new AHC. However, if you have a residential address in the EU, once you are in the EU you can obtain an EU-issued PETS passport from an EU veterinary practice, after which you will be able to use that to travel back and forth between the UK and EU as often as you like.
As with the PETS passport system, to obtain an AHC your dog or cat must have a microchip and be vaccinated for rabies. The rabies vaccination must have taken place at least 21 days prior to travel. Rabies vaccinations are valid for three years.
AHCs involve some quite time-consuming paperwork for your vet, so please contact your vet a week or two in advance of travel to arrange an AHC. You will need to take your pet to the vets for a health check within 10 days of the date of travel, at which time the AHC will be issued. That certificate will be valid for only one entry to the EU, and will be valid for only 10 days. However, the AHC will continue to be valid for four months for onward travel between EU countries, and for re-entry to Great Britain.
If you plan to travel to Northern or Southern Ireland, dogs will also have to be wormed at the vets against the tapeworm Echinococcus multilocularis within 1-5 days prior to travel.
At present, overseas travel continues to be greatly restricted because of Covid-19. If you are in a position to travel with your dog or cat, please be aware that there will be snags in this new system in the first few months, some of which may cause travellers a great deal of inconvenience and substantial expense.