This paper addresses the design of Model Predictive Control (MPC) laws to solve the trajectory-tracking problem and the path-following problem for constrained underactuated vehicles. By allowing an arbitrarily small asymptotic tracking error, we derive MPC laws where the size of the terminal set is only limited by the size of the system constraints. In fact, for the case of unconstrained inputs, the terminal set can be neglected and the resulting MPC controllers provide a global solution to the addressed constrained motion control problems. Simulation results are presented where the proposed MPC controllers are applied to 2-D and to 3-D moving vehicles.