We consider the problem of robust controller design for a class of single-input single-output nonlinear systems in strict-feedback form with structurally unknown dynamics and also with unknown virtual control coefficients. The unknown nonlinearities in the system dynamics are approximated in terms of a family of basis Functions, with the only crucial assumption made being that the parameters that characterize such a neural-network based approximation lie in some known compact sets. In this setup, we design a robust state-feedback controller under which the system output tracks a given signal arbitrarily well, and all signals in the closed-loop system remain bounded. Moreover, a relevant disturbance attenuation inequality is satisfied by the closed-loop signals. We then extend these results to the case where only the output variable is available for feedback. In this case, for tractability, the nonlinear functions in the system dynamics are restricted to depend only on the measured output variable, which results in a strict output-feedback form.