One of the most interesting properties of an impacting system is the possibility of an infinite number of impacts occurring in a finite time (such as a ball bouncing to rest on a table). Such behaviour is usually called chatter. In this paper we make a systematic study of chattering behaviour for a periodically forced, single-degree-of-freedom impact oscillator with a restitution law for each impact. We show that chatter can occur for such systems and we compute the sets of initial data which always lead to chatter. We then show how these sets determine the intricate form of the domains of attraction for various types of asymptotic periodic motion. Finally, we deduce the existence of periodic motion which includes repeated chattering behaviour and show how this motion is related to certain types of chaotic behaviour.