The problem of designing a switching observer for a finite or infinite set of nonlinear time-varying systems with unknown parameters and/or varying structure is studied in this paper. Under a compactness condition and a Lyapunov detectability assumption,(1) it is shown that a switching observer with asymptotically stable error dynamics can be constructed. Different from the variable structure control method, only a finite number of switches occur before the asymptotic stability is achieved, The Lyapunov direct method is used to construct both the switching gain and the switching index, The special case where the error dynamics becomes exponentially convergent with quadratic Lyapunov detectability is also examined to illustrate the proposed observer design procedure.