In this paper, an adaptive recurrent-neural-network (ARNN) motion control system for a biaxial motion mechanism driven by two field-oriented control permanent magnet synchronous motors (PMSMs) in the computer numerical control (CNC) machine is proposed. In the proposed ARNN control system, a RNN with accurate approximation capability is employed to approximate an unknown dynamic function, and the adaptive learning algorithms that can learn the parameters of the RNN on line are derived using Lyapunov stability theorem. Moreover, a robust controller is proposed to confront the uncertainties including approximation error, optimal parameter vectors, higher-order terms in Taylor series, external disturbances, cross-coupled interference and friction torque of the system. To relax the requirement for the value of lumped uncertainty in the robust controller, an adaptive lumped uncertainty estimation law is investigated. Using the proposed control, the position tracking performance is substantially improved and the robustness to uncertainties including cross-coupled interference and friction torque can be obtained as well. Finally, some experimental results of the tracking of various reference contours demonstrate the validity of the proposed design for practical applications.