Impulsive differential systems are an important class of mathematical models for many practical systems in physics, chemistry, biology, engineering, and information science which exhibit impulsive dynamical behaviors due to abrupt changes at certain instants during the dynamical processes. In this paper, controllability and reachability of linear piecewise constant impulsive systems are studied. Necessary and sufficient criteria for reachability and controllability are established, respectively. It is proved that reachability is equivalent to controllability for such systems under some mild conditions. Our criteria are in geometric forms, they can be transformed into algebraic type conveniently.