In this paper we study discrete-time linear systems with full or partial constraints on both input and state. It is shown that the solvability conditions of stabilization problems are closely related to important concepts, such as the right-invertibility of the constraints, the location of constraint invariant zeros and the order of constraint infinite zeros. The main results show that for right-invertible constraints the order of constrained infinite zeros cannot be greater than one in order to achieve global or semi-global stabilization. This is in contrast to the continuous-time case. Controllers for both state feedback and measurement feedback are constructed in detail. Issues regarding non-right invertible constraints are discussed as well.