Most of the previous works on the motion control of autonomous underwater vehicles (AUVs) assume that (i) the vehicle actuators are able to tolerate every level of the control signals, and (ii) the vehicle is equipped with the velocity sensors in all degrees of freedom. These assumptions are not desirable in practice. Toward this end, this paper addresses the trajectory tracking control of the underactuated AUVs with the limited torque, without the velocity measurements and under environmental disturbances in a three-dimensional space. At first, a variable transformation is introduced which helps us to derive a second-order dynamic model for underactuated AUVs. Then, a saturated tracking controller is proposed by employing the saturation functions to bound the closed-loop error variables. This technique reduces the risk of the actuators saturation by decreasing the amplitude of the generated control signals. In addition, a nonlinear saturated observer is introduced to remove the velocity sensors from the control system. The proposed controller copes with the uncertain vehicle parameters, and constant or time-varying environmental disturbances induced by the waves and ocean currents. Lyapunov's direct method is used to show the semi-global uniform ultimate boundedness of the tracking and state estimation errors. Finally, some simulation results illustrate the effectiveness of the proposed controller.