The path-following (PF) problem of an autonomous underwater vehicle (AUV) is studied, in which the path convergence is viewed as the main task while the speed profile is also taken into consideration as a secondary task. To accommodate the prioritized PF tasks, a novel multiobjective model predictive control (MPC) (MOMPC) framework is developed. Two methods, namely, weighted sum (WS) and lexicographic ordering, are investigated for solving the MOMPC PF problem. A logistic function is proposed for the WS method in an attempt to automatically select the appropriate weights. The Pontryagin minimum principle is subsequently applied for the WS-MOMPC implementation. The implicit relation between the two methods is shown, and the convergence of the solution with the MOMPC PF control algorithms is analyzed. Simulation studies on the Saab SeaEye Falcon AUV demonstrate the effectiveness of the proposed MOMPC PF control.