In this paper we present a direct adaptive control method for a class of time-varying nonlinear systems with unknown nonlinearities. We view the time-varying systems as composed of a finite number of "pieces," which are interpolated by functions that depend on a possibly exogenous scheduling variable. We assume that each piece is in strict feedback form, and show that the method yields stability of all signals in the closed-loop, as well as convergence of the state vector to a residual set around the equilibrium, whose size can be set by the choice of several design parameters. The class of systems considered here is a generalization of the class of strict feedback systems traditionally considered in the backstepping literature. We also provide design guidelines based on L/sub /spl infin// bounds on the transient.