The purpose of this chapter is two-fold: to present the most important notions of stability and to provide a geometrical interpretation of the classical Lyapunov-like methods of stability analysis. In the first section, the mathematical descriptions of the systems that will be considered in this monograph are presented. Since many results of this monograph have the same form for both continuous-time and discrete-time systems, a unified description of these classes of dynamical systems is introduced. In the second section of this chapter, the definitions of the stability properties this monograph deals with are presented. The stability properties are classified into three groups depending on the characteristics of disturbances applied on the system or on their effect on the mathematical modeling of the system. In the third section, based on the link between positive definite functions and the notion of distance, a geometrical interpretation of Lyapunov’s direct method is given.