They are easy going, laid back, calm, clean, and very sweet. Easy to housebreak, enjoyable and barely there house companions... they can thrive in the smallest apartment, yet they love long walks in the woods and can be great in the country, too.
I've had Whippets for years, and all of mine could be trusted with cats, etc. Some will kill squirrels, but I don't find that offensive
As mentioned, they don't bark either, which is grand.
Mine are also all excellent off leash, when raised from puppyhood. They're just not that interested in strangers, etc, so they tend to stick to their owners (unlike my overly friendly Goldens, which run up to anyone they see without extensive training). They also tend towards a bit of separation/over attachment to one person. This makes them very intuitive and bonded, but it can be annoying- most suffer deeply if boarded in kennels, for example. They cannot be kept outside humanely, and they often dislike crates. They do tend to favor one person. Provided that person is me, I love it...
but it can be irritating to other family members if your Whippet really only is animated with and interested in his one special soulmate.
BTW while they do need exercise, they are very lazy dogs indoors. They require MUCH LESS exercise than a Golden, Lab, or most other popular medium sized to large dogs. That doesn't mean you shouldn't encourage running and playing and take your Whippet for walks, though- you must!
It reduces boredom and keeps them nice and fit.