Generally, all airline companies allow dogs and other pets transportation aboard, however each airline has its own rules, so it is always a good idea to plan ahead and get relevant information.
You must inform your airline that you are traveling with a pet before actually making the reservation.
If you are traveling to a foreign destination you must know in advance all relevant regulations: some Countries require your pet stay in a quarantine period (sometimes 3 months).
As a general rule your pet must be vaccinated against major disease, and it is always advisable to bring along all vaccination documentation and your veterinary certificate of good heath.
All airlines allow you to bring a small dog inside the aircraft cabin, provided they have a waterproof cage (maximum dimensions are 47x25x32 cm with 10 Kg of max weight).
In case your pet and the cage weigh more 10 kg altogether, the animal will travel in the cargo area, in special cages provided by the airlines.
There are exceptions for guiding dogs for visually impaired passengers, such dogs must be leashed and with a muzzle. Guiding dogs travel free of charge, however you must let the airline know prior to reservation. Passenger and dog will board the aircraft before other passengers and will travel side by side.
Birds, fish, frogs and turtles may be brought to Italy only with their Country of origin certificate.
Endangered species (parrots, sea turtles, exotic fish, etc.) must have their CITES certificate (export authorization).