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A robust adaptive controller for a nonholonomic mobile robot with unknown kinematic and dynamic parameters is proposed. A kinematic controller whose output is the input of the relevant dynamic controller is provided by using the concept of backstepping. An adaptive algorithm is developed in the kinematic controller to approximate the unknown kinematic parameters, and a simple single-layer neural network is used to express the highly nonlinear robot dynamics in terms of the known and unknown parameters. In order to attenuate the effects of the uncertainties and disturbances on tracking performance, a sliding mode control term is added to the dynamic controller. In the deterministic design of feedback controllers for the uncertain dynamic systems, upper bounds on the norm of the uncertainties are an important clue to guarantee the stability of the closed-loop system. However, sometimes these upper bounds may not be easily obtained because of the complexity of the structure of the uncertainties. Thereby, simple adaptation laws are proposed to approximate upper bounds on the norm of the uncertainties to address this problem. The stability of the proposed control system is shown through the Lyapunov method. Lastly, a design example for a mobile robot with two actuated wheels is provided and the feasibility of the controller is demonstrated by numerical simulations.