They consider the following animals as okay to fly in the cabin:
Dogs
Cats
Birds (excluding birds of prey)
Fish
Tortoises
Rodents such as hamsters, guinea pigs and small rabbits
The following creatures must be transported in the hold:
Amphibians
Insects
Crustaceans
Reptiles
Other rodents than those mentioned above
Ferrets
Livestock not allowed in either of these categories include:
Animals unaccompanied by their owners
Commercial animal shipments
Protected endangered species
Adult carnivores not categorised as "pets"
Laboratory animals
Pregnant animals
I know most of you will be simply looking to transport your dog, cat, or maybe a hamster or rabbit. If you want your dog to be with you in the cabin, it must be small enough for it (inside its cage or travelling bag) to fit under the seat in front. Any bigger pets will have to be put into the hold.
Regarding hold animals, any passenger can transport up to five animals at one time.
There can be up to three animals in one container, but these must be of the same species.
Maximum weight for each container, plus its contents, must not exceed eight kilograms.
Potentially dangerous dogs must be muzzled. These include the breeds:
Pit bull terrier
Staffordshire bull terrier
Rottweiler
The containers must comply with the conditions outlined on the main "pets" page.
There are a whole host of other regulations, mostly concerned with vaccinations and "pets' passports", as outlined on the "pets" page, too, but, if you're thinking of using Air Europa, you'd do well to refer to their website.