Of the world's seven continents, only one has no resident cats of any kind whatsoever: Antarctica. Domestic cats have come there with expeditions and gone with them, too, unable to stay on their own -- lethal climate, no prey. All others have cats, native or introduced.
Africa
Africa claims both large and small cats, from the mighty lion to the tiny black-footed cat, a possible ancestor of Felis domesticus. In between these extremes are the leopard, the cheetah, the serval, the caracal and numerous other felines, as well as the so-called civet cat, which is not a cat at all, but a viverrid.
Asia
Asia is home to the largest of all the cats, the Siberian tiger, and the smallest of cats, the rusty-spotted cat of India and Sri Lanka. In the middle are the leopard, the snow leopard, the clouded leopard and on down to the small wild cats such as the odd-looking flat-headed cat and the fishing cat, the only wild cat known to fish on a regular basis.
Europe
The lion and the leopard no longer roam in Europe, and the largest remaining wild feline is the Scottish wildcat. The Eurasian lynx is larger, but is now found mostly in Scandinavia and eastern Russia, as far from civilization as he can get.
North America
The largest cat species in North America is the mountain lion, also known as a cougar or puma, but the largest individual cat in the world, a liger named Hercules, lives in Florida. Hercules is a hybrid resulting from the mating of a male lion and a female tiger in captivity, so he is not a species, but he is certainly a cat. Smaller wild cats of North America range from the bobcat, a cousin of the lynx, to the jaguarundi of Mexico and South Texas.
South America
The burly jaguar stalks the rainforest of Central and South America along with his tiny cousins, the oncilla and the kodkod, while the shy puma ranges the pampas and the Andean cat hunts chinchilla in the mountains.
Australia
Until humans arrived, Australia had no cats at all. The ecological niche they would have occupied was filled by equivalent predatory carnivorous species, including the marsupial cat and the thylacine or "Tasmanian tiger." Today, many native species are threatened by an exploding population of feral domestic cats. There are some wild tales of big cats prowling the outback, but if they're out there, they're probably escaped exotics.
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