For a given continuous-time switched system, its stability is determined by (i) the structure of the system; (ii) the admissible switching set; and (iii) the duration time in each subsystem. Both (ii) and (iii) are stability characterizations of switching dynamics for continuous-time switched systems. In this paper, we introduce a new type of characteristic exponent, the Liao-type exponent, which is computable, to capture the stability feature of continuous-time switched systems. We obtain two criteria of asymptotic exponential stability for linear and nonlinear cases by estimating the corresponding Liao-type exponents, respectively. The proposed approach is constructive and the obtained results allow existence of switching constraints. Several examples are provided to illustrate theoretical outcomes of the paper.