This paper represents a study of the key concepts of attracting sets (submanifolds) and partially stable systems aimed at establishing relations between the local properties and the unification of the methodologies of their analysis. The use of recent results of stability theory and techniques of geometric control enables one to generalize Lyapunov-like sufficient conditions and propose simplified solutions for the problems of set attractivity and partial stability. (C) 2003 Elsevier Ltd. All rights reserved.