This paper is concerned with neural-network (NN)-based adaptive control schemes for a class of nonlinear system which includes a finite Volterra series system and a Wiener system. First, introducing a new kind of dynamic neural network which consists of Laguerre filters and memoryless nonlinear elements, a model reference adaptive control (MRAC) scheme is presented for the nonlinear systems. In the proposed MRAC system adopting overparameterization and a robust adaptive algorithm, the boundedness of the estimated parameters is assured under some conditions. Second, an adaptive linearization scheme for Wiener systems with nonlinearity in their output part is realized by using a kind of functional-link network. It is shown that the obtained controller has a structure similar to the MRAC and then the boundedness of the estimated parameters as well as that of all the signals in the closed loop are also ensured. Finally, the effectiveness of the proposed schemes is illustrated through numerical simulations.