This paper is concerned with the target tracking of underactuated autonomous surface vehicles with unknown dynamics and limited control torques. The velocity of the target is unknown, and only the measurements of line-of-sight range and angle are obtained. First, a kinematic control law is designed based on an extended state observer, which is utilized to estimate the uncertain target dynamics due to the unknown velocities. Next, an estimation model based on a single-hidden-layer neural network is developed to approximate the unknown follower dynamics induced by uncertain model parameters, unmodeled dynamics, and environmental disturbances. A bounded control law is designed based on the neural estimation model and a saturated function. The salient feature of the proposed controller is twofold. First, only the measured line-of-sight range and angle are used, and the velocity information of the target is not required. Second, the control torques are bounded with the bounds known as a priori. The input-to-state stability of the closed-loop system is analyzed via cascade theory. Simulations illustrate the effectiveness of the proposed bounded controller for tracking a moving target.