In this paper, we develop some general results on the properties of the reachable sets for right invariant bilinear systems with state varying on compact Lie groups. The main results consist of a characterization of the set of states reachable in arbitrary time from the identity of the group. This, under suitable assumptions, is proved to be a Lie subgroup of the underlying Lie group. We apply these results to the analysis of the controllability of particles with spin. For these systems we also obtain estimates of the time after which every state is reachable from the identity. The results are motivated by the problem of controlling a two-level quantum system in implementations of quantum computers.