The presence of constraints in the on-line optimization problem solved by Model Predictive Control algorithms results in a nonlinear control system, even if the plant and model dynamics are linear. This is the case both for physical constraints, like saturation constraints, as well for performance or safety constraints on outputs or other variables of the process. This paper discusses how constraints affect the stability properties of the closed-loop nonlinear system. In particular we concentrate on presenting a formulation that allows one to relate hard as well as soft constraints to stability. The degree of softening can be determined to guarantee stability.