In this paper, a stable robust adaptive control approach is presented for a class of unknown nonlinear systems in the strict-feedback form with disturbances. The key assumption is that neural network approximation errors and external disturbances satisfy certain bounding conditions. By combining neural network technique with backstepping method and introducing a special type of Lyapunov functions, the controller singularity problem is avoided perfectly. As the estimates of unknown neural network approximation error bound and external disturbances bound are adjusted adaptively, the robustness of the closed-loop system is improved and the application scope of nonlinear systems is extended. The overall neural network control systems can guarantee that all the signals of the closed-loop system are uniformly ultimately bounded and the tracking error converges to a small neighborhood of zero by suitably choosing the design parameters. The feasibility of the control approach is demonstrated through simulation results.