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. (C) 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.