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This study addresses the problem of position-tracking control of underactuated autonomous underwater vehicles (AUVs) in the presence of unknown ocean currents in a horizontal plane. A position-tracking controller and a current observer are presented based on the Lyapunov stability theory by using the backstepping technique, respectively. The final controlled system, which arises from putting together the position-tracking controller and the observer, is proved to be globally K-exponentially stable by stability criteria for cascade system. The trajectories used for the illustration of the proposed control schemes are a circle with constant velocity and a sinusoidal curve that requires time-varying velocity. In order to demonstrate the practicability of the proposed controller, the surge force is assumed to be non-negative, and control input saturations are considered. Simulation results are presented to demonstrate the effectiveness of the proposed control schemes.