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This paper presents a robust adaptive control design method for a class of multiple-input-multiple-output uncertain nonlinear systems in the presence of parametric and nonparametric uncertainties and bounded disturbances. Using the approximation properties of the unknown continuous nonlinearities and the adaptive bounding technique, the developed controller achieves asymptotic convergence of the tracking error to zero, while ensuring boundedness of parameter estimation errors. The algorithm does not assume the knowledge of any bound on the unknown quantities in designing the controller. It is based on an integral technique involving the filtered tracking error and produces a continuous control. Theoretical developments are illustrated via simulation results.