The paper deals with the review of results obtained in stability investigation of hybrid dynamical systems. Systems of this type are capable of exhibiting simultaneously several kinds of dynamic behavior in different parts of the system (e.g. continuous-time dynamics, discrete-time dynamics, logic commands, discrete events, jump phenomena, etc.). In terms of one measure

$d(p(t,a,{t}_{0}),M)$ (the distance from the trajectory

$p(t,\xb7)$ to the invariant set

$M$) the definitions of various types of stability are introduced which were earlier formulated in classical theory of stability. Besides, a “generalized” time is used. The results presented are applied in the monograph [

*A. N. Michel, K. Wang* and

*Bo Hu*, “Qualitative theory of dynamical systems. The role of stability preserving mappings”. Marcel Dekker, Inc., New York (2001)].