In this paper, the model matching problem of nonlinear single-input single-output discrete-time systems is considered. Both feedforward and feedback solutions are given. The problem is studied within the transfer function approach in which the system is described by the quotient of two polynomials from the non-commutative ring of skew polynomials. In general, the feedforward solution does not always exist, therefore the subclass of nonlinear control systems is specified for which the solution exists. The feedback solution exists always. Additionally, the conditions for the existence of the proper compensator are provided. The developed theory and algorithm are illustrated of several examples.