We present a solution to the following reduction problem for asymptotic stability of closed sets in nonlinear systems. Given two closed, positively invariant subsets of the state space of a nonlinear system, Gamma(1) subset of Gamma(2), assuming that Gamma(1) is asymptotically stable relative to Gamma(2), find conditions under which Gamma(1) is asymptotically stable. We also investigate analogous reduction problems for stability and attractivity. We illustrate the implications of our results on the stability of sets for cascade-connected systems and on a hierarchical control design problem. For upper triangular control systems, we present a reduction-based backstepping technique that does not require the knowledge of a Lyapunov function, and mitigates the problem of controller complexity arising in classical backstepping design. (C) 2012 Elsevier Ltd. All rights reserved.