It varies. As a rule, cats will leave mature chickens alone but kill chicks. However, some people do have problems with cats killing their chickens -- usually feral cats.
I think it depends some on the personality of the cat. My chickens are in a fenced yard. My cat watches and crouches outside the yard, and the chickens get a bit nervous and move away from the fence. I used to have a different cat and unfenced chickens. The cat would hide in the weeds and jump out at the chickens, but never touch them. After a while the chickens ignored the cat. In many cases a chicken pecks the cat and after that the cat keeps his distance. I've also seen pics here of cats sleeping in nest boxes.