This paper presents a solution to the problem of path following control for autonomous marine vehicles (AMVs) subject to input constraints and constant ocean current disturbances. We propose two nonlinear control strategies: the first is obtained by using a Lyapunov-based design method, while the second is developed by adopting a Model Predictive Control (MPC) framework. We show that, with the proposed control strategies, the path-following error is globally asymptotically stable (GAS). Simulations with a kinematic model of the vehicle support the theoretical results. Simulations with a realistic model of the Medusa class of AMVs show the robustness of the proposed control strategies.