Hybrid cat breeds are popular with a certain crowd, but some are banned in certain states in the U.S. and other parts of the world. These pet cats are a mix between a domestic cat, and a wild cat species.
The two best known cats of this type are probably the Bengal and the Savannah, but others include Chausie (Stone Cougars), and Safari Cats. The CFA doesn't recognize hybrids, but other Cat Fancy organizations, such as TICA, do.
Holy Cross-Breeds Batman
Below are some of the hybrid cat breeds that exist. If you're interested in one of these cats, keep in mind that they will likely be expensive. They also are not allowed by law in certain areas (check your local ordinances), and they can be a handful to care for.
There's a certain "designer" pet aspect to these kitties, largely due to the combination of the high price and the exotic nature of the animals. Unfortunately this results in some adoptions being done without regard to the commitment required to caring for a cat.
Bengal cats
Bengal cats are a cross between Asian leopard cats and domestic cats. Bengal cat videos are very popular on Youtube, with their owners either showing off their jumping and agility skills, or their love of playing in the bathtub.
Savannah cats
Savannah cats cross the African serval with a domestic cat. Servals are quite a bit larger than house cats, and the resulting Savannah is a big kitty. They're long and lean, with long necks and legs.
The Savannah cat breed first appeared in the late 1980s, and were accepted by TICA in 2001. They can be shown in the Advanced New Breed Class.
Chausie
The Chausie (or Stone Cougar) are a Jungle cat and domestic cat cross. This cat has a bit of a cougar-like look.
Chausies were first bred in the late 1960s, and are TICA registered and can be shown in the Advanced New Breed class.
Toyger cats
Toyger cats are domestic cats bred to look like little tigers. The breed was started by Judy Sugden, by crossing a Bengal with a striped domestic shorthair.
Toyger breeders are shooting for a coat that looks as tiger-like as possible, with dark stripes on an orange/tan background. The long random stripes should be more dramatic looking than a typical tabby cat, and there should be no spotting.
Safari
The Safari cat is a hybrid cat breed consisting of the South American Geoffroy's cat and the domestic cat, first bred in the 1970s. They are said to be difficult to breed, and are rare, even for hybrids.
Serengeti
In the mid 1990's, Karen Sausman created the Serengeti cat with the African serval in mind. Unlike the Savannah, however, the Serengeti breed has no serval genes and was created by crossing Bengal cats with Oriental Shorthairs. Like the other hybrids, they are not recognized by the CFA, but are TICA registered and can be shown as a preliminary new breed.
Pixie Bob
Pixie Bob cats are sometimes said to have come from wild bobcat lineage. Recent research into domestic feline genetics has yielded some surprises, and revealed some interesting history of the house cat.
I haven't been able to find, however, any documentation that shows any evidence of bobcat genes in the line. It makes for a good breeding legend, though!