The paper deals with stability of solutions of nonlinear dynamical systems with complex changing structure of connections. The main results of stability analysis by means of nonlinear approximation are presented. Important aspects of applying the theory of differential inequalities and the method of comparison to stability analysis are described. The question of construction of auxiliary comparison systems for the considered dynamical systems is studied. Stability criteria for those comparison systems are formulated. Methods of decomposition and aggregation of complex systems are considered. The switches influence on stability is estimated. Some directions of classical results development, relevant in recent years, are shown. In particular, these results are applied to the problem of absolute stability, the modeling of population interactions in biological communities, and the stability analysis of the equilibrium positions of mechanical systems.