This book is an introduction to perturbation methods, a systematic way of constructing approximate solutions to problems that are otherwise intractable. The methods rely on the presence of a small parameter in the problem, which is common in applications. Perturbation methods are a cornerstone of applied mathematics, along with scientific computing, and they are used to analyze nonlinear, inhomogeneous, and multidimensional problems. The goal of perturbation methods is to provide a reasonably accurate expression for the solution, which can lead to a better understanding of the physics of the problem and more efficient numerical procedures. The book covers a wide range of topics, including asymptotic approximations, matched asymptotic expansions, multiple scales, the WKB method, homogenization, bifurcation and stability, and more. The first chapter introduces the fundamental ideas of asymptotic approximations, including their use in solving transcendental and differential equations. The second chapter discusses matched asymptotic expansions for problems with layers. Chapter 3 describes methods for dealing with problems with multiple time scales. Chapter 4 develops the WKB method for linear singular perturbation problems, while Chapter 5 discusses methods for dealing with materials with disparate spatial scales. The last chapter examines multiple solutions and stability. The mathematical prerequisites include a basic background in differential equations and advanced calculus. The chapters are written so that the first sections are either elementary or intermediate, while the later sections are more advanced. The ideas developed in each chapter are applied to a variety of problems, including ordinary differential equations, partial differential equations, and difference equations. The exercises vary in complexity and include problems from the research literature. Some solutions to the exercises are available on the author's home page. The book has been revised for the second edition, with many sections edited, some completely revised, and new material added. Two appendices have been added, one on solving difference equations and the other on delay equations. The references have been updated, new exercises added, and some original exercises modified. The book also includes an errata sheet and answers to some of the exercises. The code used for the figures is available on the author's home page, as well as videos showing some of the solutions of the time-dependent problems solved in the book.This book is an introduction to perturbation methods, a systematic way of constructing approximate solutions to problems that are otherwise intractable. The methods rely on the presence of a small parameter in the problem, which is common in applications. Perturbation methods are a cornerstone of applied mathematics, along with scientific computing, and they are used to analyze nonlinear, inhomogeneous, and multidimensional problems. The goal of perturbation methods is to provide a reasonably accurate expression for the solution, which can lead to a better understanding of the physics of the problem and more efficient numerical procedures. The book covers a wide range of topics, including asymptotic approximations, matched asymptotic expansions, multiple scales, the WKB method, homogenization, bifurcation and stability, and more. The first chapter introduces the fundamental ideas of asymptotic approximations, including their use in solving transcendental and differential equations. The second chapter discusses matched asymptotic expansions for problems with layers. Chapter 3 describes methods for dealing with problems with multiple time scales. Chapter 4 develops the WKB method for linear singular perturbation problems, while Chapter 5 discusses methods for dealing with materials with disparate spatial scales. The last chapter examines multiple solutions and stability. The mathematical prerequisites include a basic background in differential equations and advanced calculus. The chapters are written so that the first sections are either elementary or intermediate, while the later sections are more advanced. The ideas developed in each chapter are applied to a variety of problems, including ordinary differential equations, partial differential equations, and difference equations. The exercises vary in complexity and include problems from the research literature. Some solutions to the exercises are available on the author's home page. The book has been revised for the second edition, with many sections edited, some completely revised, and new material added. Two appendices have been added, one on solving difference equations and the other on delay equations. The references have been updated, new exercises added, and some original exercises modified. The book also includes an errata sheet and answers to some of the exercises. The code used for the figures is available on the author's home page, as well as videos showing some of the solutions of the time-dependent problems solved in the book.