This paper is concerned with the study of, both local and global, uniform asymptotic stability for general nonlinear and time-varying switched systems. Two concepts of Lyapunov functions are introduced and used to establish uniform Lyapunov stability and uniform global stability. With the help of output functions, an almost bounded output energy condition and a persistent excitation (PE) condition related to output functions are then proposed and employed to guarantee uniform (global) asymptotic stability. Based on this result, a generalized version of Krasovskii-LaSalle theorem in time-varying switched systems is proposed. Moreover, it is shown that the proposed PE condition can be verified by checking a zero-state observability condition for switched systems with persistent dwell-time. Particularly, our results can be used in the case where only some parts of the overall switched system are stable. It is shown that several existing results in past literature can be covered as special cases using the proposed criteria.