This book provides an introduction to numerical continuation methods, which are two techniques used for solving nonlinear systems of equations. The first method, called the predictor-corrector or pseudo arc-length continuation method, is based on historical methods used by engineers and scientists to improve convergence. The second method, known as the simplicial or piecewise linear method, has roots in the Lemke-Howson algorithm for solving nonlinear complementarity problems. Both methods are referred to as continuation methods due to their similarities in principles and implementations. The book discusses the predictor-corrector methods in chapters 3–10 and the piecewise linear methods in chapters 12–16. Chapter 11 bridges the two approaches by covering applications where either or both methods may be used. The authors emphasize that while the methods appear distinct, they share many common features and are based on similar principles. The book aims to provide an accessible introduction to these methods for scientific workers and students. The authors use pseudocode in Pascal syntax to describe algorithms, as it is considered the clearest way to present the material. They also provide FORTRAN programs and numerical examples in the appendix, which are primarily for illustration. The authors acknowledge that these programs are not perfected and encourage readers to adapt them for their specific applications. The book includes a detailed bibliography and a table of contents covering various topics, including the basic principles of continuation methods, convergence of Euler-Newton-like methods, steplength adaptations, predictor-corrector methods using updating, bifurcation detection, calculating special points of the solution curve, large-scale problems, numerically implementable existence proofs, PL continuation methods, PL homotopy algorithms, general PL algorithms on PL manifolds, approximating implicitly defined manifolds, and update methods and their numerical stability. The authors also mention that the codes for the programs will be available via email for a limited time. They thank their colleagues for their contributions and acknowledge the help and support they received during the preparation of the book. The book is intended for readers with a background in elementary analysis and linear algebra, and some knowledge of numerical analysis may be helpful. The authors hope that the book will help readers understand the numerical aspects of both methods and choose the most appropriate one for their tasks.This book provides an introduction to numerical continuation methods, which are two techniques used for solving nonlinear systems of equations. The first method, called the predictor-corrector or pseudo arc-length continuation method, is based on historical methods used by engineers and scientists to improve convergence. The second method, known as the simplicial or piecewise linear method, has roots in the Lemke-Howson algorithm for solving nonlinear complementarity problems. Both methods are referred to as continuation methods due to their similarities in principles and implementations. The book discusses the predictor-corrector methods in chapters 3–10 and the piecewise linear methods in chapters 12–16. Chapter 11 bridges the two approaches by covering applications where either or both methods may be used. The authors emphasize that while the methods appear distinct, they share many common features and are based on similar principles. The book aims to provide an accessible introduction to these methods for scientific workers and students. The authors use pseudocode in Pascal syntax to describe algorithms, as it is considered the clearest way to present the material. They also provide FORTRAN programs and numerical examples in the appendix, which are primarily for illustration. The authors acknowledge that these programs are not perfected and encourage readers to adapt them for their specific applications. The book includes a detailed bibliography and a table of contents covering various topics, including the basic principles of continuation methods, convergence of Euler-Newton-like methods, steplength adaptations, predictor-corrector methods using updating, bifurcation detection, calculating special points of the solution curve, large-scale problems, numerically implementable existence proofs, PL continuation methods, PL homotopy algorithms, general PL algorithms on PL manifolds, approximating implicitly defined manifolds, and update methods and their numerical stability. The authors also mention that the codes for the programs will be available via email for a limited time. They thank their colleagues for their contributions and acknowledge the help and support they received during the preparation of the book. The book is intended for readers with a background in elementary analysis and linear algebra, and some knowledge of numerical analysis may be helpful. The authors hope that the book will help readers understand the numerical aspects of both methods and choose the most appropriate one for their tasks.