This paper investigates the problem of designing a compensating control law for a square invertible nonlinear plant so that the response of the closed-loop system asymptotically matches that of a prescribed, driven, nonlinear model. A set of necessary conditions for achieving asymptotic model matching is established. One of these involves the stability properties for a subdynamics which is common to every model matching closed loop. This subdynamics, called the "fixed dynamics," is intrinsically characterized. Based on these results, a new set of sufficient conditions for achieving asymptotic model matching is given. The interrelation between the minimum-phase and vector relative degree properties of a plant and a matchable model are studied.