A significant number of Model Predictive Control algorithms solve on-line an appropriate optimization problem and do so at every sampling point. The major attraction of such algorithms, like the Quadratic Dynamic Matrix Control (QDMC), lies in the fact that they can handle hard constraints on the inputs (manipulated variables) and outputs of a process. The presence of such constraints results in an on-line optimization problem that produces a nonlinear controller, even when the plant and model dynamics are assumed linear. This paper provides a theoretical framework within which the stability and performance properties of such algorithms can be studied. Necessary and/or sufficient conditions for nominal and robust stability are derived and two examples are used to demonstrate their effectiveness.