In July a new airline called “Pet Airways” will launch that is just for dogs and cats:

All animals that fly will be called “pawsengers” and have a pet attendant to ease their travel, says airline co-founder Alysa Binder. Pet Airways will start out in five cities — Los Angeles, New York, Washington, Chicago and Denver. The animals will fly on Beechcraft planes, constructed for 19 human passengers, Binder tells Renee Montagne. All the people comforts have been torn out and reconfigured with pet carriers. There are sections for cats and dogs, but there’s no first class. Source

Fares start at $149 – ouch! but do include food/water.