A level curve approach is introduced to design global path-following controllers for an underactuated surface ship. The approach is based on the observation: if the position of the ship satisfies the equation of the reference path, then the ship will be on the path. Thus, the controllers are designed based on Lyapunov's direct and backstepping methods to force the position of the ship to satisfy the equation of the path and to move along the path tangentially. The approach does not require computation of the position from the ship to the path. Weak and strong nonlinear Lyapunov functions are introduced in the control design to overcome difficulties caused by underactuation and to guarantee boundedness of the sway velocity. Simulations are included to illustrate the effectiveness of the proposed results.