This chapter introduces conditions that guarantee whenever the system starts near an equilibrium state, it remains near it, perhaps even converging to the equilibrium state as the time increases. These kinds of stability are called the Lyapunov-stability and asymptotical stability, respectively. The chapter introduces the Lyapunov stability theory to examine Lyapunov stability and asymptotical stability of linear and nonlinear systems. It then introduces and investigates external stability for linear systems, when we finds conditions that with zero initial state a bounded input always evokes a bounded output. The chapter analyzes the stability of linear time-invariant systems given in state-space form. It discusses the methods based on transfer functions. The chapter illustrates the applications of the stability analysis of dynamic systems via particular systems arising in engineering and social sciences.