There are playable classes such as the Death Knight and Sky Guard that have mounts. At above lvl 50 Death Knights can fly Mount that classify (name wise) under this group of creatures. At and around Lvl 50 Sky Guard can fly Dragons like The Two Headed Wyverns (young and old).
NOTE: Death Knights and Sky Guard are not faster travelers just because they have mounts. It costs the same amount of Action Points that it would an unmounted class. However there are Advantages that you can buy to help you travel faster.