Whippets were bred to course and kill rabbits...small furry prey that runs. Though most whippets are not used for that purpose today, they are still bred with that instinct in mind, and the ones who are coursed and raced are bred specifically for high prey drive. Very few whippets are born without it. Even if your dog is raised with your cat and loves and plays gently with it, if he ever sees the cat running outside, it won't be his beloved Muffy, but prey. And whippets are "kill hounds." They don't play with their prey or bring it back alive. By the way he realizes that it was Muffy he was chasing, it will be too late.
Some owners do keep cats and whippets together successfully, but our experience has not been happy. The prey drive instinct is "hard-wired" into these dogs and even the best training cannot be guaranteed to control basic instinct. Not all whippets will kill a running cat, but most of them will. If you do have a whippet and a cat, we suggest never leaving them alone together unsupervised, and certainly you should never put the two outside at the same time.
*Note from Johannah: Derek and I have kept whippets and cats together successfully for 15 years. We have a few rules however -- the cats are NEVER allowed outside, and the dogs are NEVER left loose in the house unsupervised with the cats.
The same advice applies to small dogs, especially very furry ones. Whippets play roughly with other dogs, and part of their play is to chase and grab. A small, fast-moving dog is not really a good companion.
Along the same line, if your whippet has a yard, he's liable to kill something. Birds, squirrels, opossums...anything that wanders in is fair game. This can be traumatic for some people. If your own nature is too sensitive to be able to love a dog that may kill wildlife, consider a collie instead of a whippet. Some whippets have a stronger prey drive than others, and some whippets may go their entire lives without killing anything.