This paper presents a solution to the problem of trajectory tracking control for autonomous surface craft (ASC) in the presence of ocean currents. The proposed solution is rooted in nonlinear model predictive control (NMPC) techniques and addresses explicitly state and input constraints. Whereas state saturation constraints are added to the underlying optimization cost functional as penalties, input saturation constraints are made intrinsic to the nonlinear model used in the optimization problem, thus reducing the computational burden of the resulting NMPC algorithm. Simulation results, obtained with a nonlinear dynamic model of a prototype ASC, show that the NMPC strategy adopted yields good performance in the presence of constant currents. Experimental results are also provided to validate the real-time implementation of the NMPC techniques for ASCs.