We consider adaptive output feedback control of uncertain multi-input multi-output nonlinear systems, in which both the dynamics and the dimension of the regulated plant may be unknown, but knowledge of vector relative degree is required. Given smooth reference trajectories, the problem is to design controllers that force the system measurements to track them with bounded errors. We propose a linear observer for the output tracking error vector and a Single Hidden Layer (SHL) Neural Network (NN) to cancel the modelling errors. Ultimate boundedness of the error signals is shown through Lyapunov's direct method. Simulations of a fourth order two-input two-output nonlinear system illustrate the theoretical results.