In the last decade several results on cascaded autonomous systems have appeared in the literature. The sufficient conditions for the stability and stabilizability of these systems are often related to certain growth rate conditions on the functions which define the dynamics of the system. In this paper we analyze three complementary classes of nonlinear time-varying systems according to these growth rates. For each case. we give further results to guarantee uniform global asymptotic stability of the cascade and relate our contributions to input-to-state stability and other growth rate conditions previously reported. The advantage of this type of analysis is that it is often easier to verify such conditions than to find a Lyapunov function with a negative-definite derivative.