This paper describes a novel learning approach to asymptotic state tracking in a class of nonlinear systems. The tracking problem considered here concerns the case when the tracking-error dynamics are described by a set of time-varying nonlinear differential equations, which are periodic in time with a known period. Our iterative update scheme is based on the specific property that the learning system tends to oscillate in steady state. In fact, our approach extends in a very natural manner the idea of the well-known iterative learning control for the case of finite-time tracking problems to the case of infinite-time asymptotic tracking problems. The best advantage of the proposed learning approach is that it is computationally simple and does not require one to solve any complicated equations based on full system dynamics. We explore the conditions under which a periodic nonlinear system exhibits a steady-state oscillation. Our work also can be viewed to provide a learning-based solution to the input-state inversion problem. The generality and practicality of our work is demonstrated through rigorous performance analysis and simulation using a robot manipulator.