This paper addresses the development of stabilizing state and output feedback model predictive control (MPC) algorithms for constrained continuous-time nonlinear systems with discrete observations. Moreover, we propose a nonlinear observer structure for this class of systems and derive sufficient conditions under which this observer provides asymptotically convergent estimates. The MPC scheme proposed here consists of a basic finite horizon nonlinear MPC technique with the introduction of an additional state constraint, which has been called a contractive constraint, The resulting MPC scheme has been denoted contractive MPC (CNTMPC), This is a Lyapunov-based approach in which a Lyapunov function chosen a priori is decreased, not continuously, but discretely; it is allowed to increase at other times (between prediction horizons). We will show in this work that the implementation of this additional constraint into the on-line optimization makes it possible to prove strong nominal stability properties of the closed-loop system. In the absence of disturbances, it can be shown that the presence of the contractive constraint renders the closed-loop system exponentially stable in the state feedback Ease and uniformly asymptotically stable in the output feedback case.