This book presents a comprehensive survey of mathematical definitions of internal and external stability in nonlinear control systems, focusing on the use of Lyapunov functions. It discusses the characterization of stability through Lyapunov functions and the design of feedback laws to achieve improved stability. The book begins by considering continuous-time, finite-dimensional systems described by ordinary differential equations, and the associated unforced system. It then explores the distinction between internal and external behaviors of systems, with external behavior being influenced by inputs. The text emphasizes the importance of Lyapunov functions in analyzing and stabilizing nonlinear systems, and introduces the concept of stabilizability, where feedback laws are used to improve system performance. The book also addresses the limitations of smooth systems and introduces nonsmooth functions and differential equations with discontinuous right-hand sides. It discusses the existence of Lyapunov functions, their properties, and their applications to stability and stabilization. The text covers time-invariant and time-varying systems, and presents direct and converse theorems on stability and asymptotic stability. It also explores the relationship between Lyapunov functions and trajectory decay, as well as the use of nonsmooth analysis in the study of discontinuous Lyapunov functions. The book concludes with a review of tools from nonsmooth analysis useful for analyzing non-differentiable systems. The content is organized into chapters covering differential equations, time-invariant systems, time-varying systems, differential inclusions, and additional properties of Lyapunov functions.This book presents a comprehensive survey of mathematical definitions of internal and external stability in nonlinear control systems, focusing on the use of Lyapunov functions. It discusses the characterization of stability through Lyapunov functions and the design of feedback laws to achieve improved stability. The book begins by considering continuous-time, finite-dimensional systems described by ordinary differential equations, and the associated unforced system. It then explores the distinction between internal and external behaviors of systems, with external behavior being influenced by inputs. The text emphasizes the importance of Lyapunov functions in analyzing and stabilizing nonlinear systems, and introduces the concept of stabilizability, where feedback laws are used to improve system performance. The book also addresses the limitations of smooth systems and introduces nonsmooth functions and differential equations with discontinuous right-hand sides. It discusses the existence of Lyapunov functions, their properties, and their applications to stability and stabilization. The text covers time-invariant and time-varying systems, and presents direct and converse theorems on stability and asymptotic stability. It also explores the relationship between Lyapunov functions and trajectory decay, as well as the use of nonsmooth analysis in the study of discontinuous Lyapunov functions. The book concludes with a review of tools from nonsmooth analysis useful for analyzing non-differentiable systems. The content is organized into chapters covering differential equations, time-invariant systems, time-varying systems, differential inclusions, and additional properties of Lyapunov functions.