Paleantologists call them "Gorgonopsids" and believe that they've been extinct for millions of years. Isn't it fun when science is wrong?
There are two known species of Questing Beast. They may both be extinct; we really don't know, and given the tendency of Questing Beasts to eat anyone who comes remotely near them, we're really not in any hurry to find out. The North American variety at least rattles when it's getting ready to launch itself at your head, which is much more polite than its European counterpart.
African and Indian Questing Beasts have been theorized from the fossil record. We'd really rather not think about that right now.