Basic Material -- Approximation of Integrals -- Boundary Layer Behaviour -- Two-Point Boundary Value Problems -- Nonlinear Boundary Value Problems -- Elliptic Boundary Value Problems -- Boundary Layers in Time -- Evolution Equations with Boundary Layers -- The Continuation Method -- Averaging and Timescales -- Advanced Averaging -- Averaging for Evolution Equations -- Wave Equations on Unbounded Domains.Perturbation theory, one of the most intriguing and essential topics in mathematics, and its applications to the natural and engineering sciences is the main focus of this workbook. In a systematic introductory manner, this unique book deliniates boundary layer theory for ordinary and partial differential equations, multi-timescale phenomena for nonlinear oscillations, diffusion and nonlinear wave equations. The book provides analysis of simple examples in the context of the general theory, as well as a final discussion of the more advanced problems. Precise estimates and excursions into the theoretical background makes this workbook valuable to both the applied sciences and mathematics fields. As a bonus in its last chapter the book includes a collection of rare and useful pieces of literature, such as the summary of the Perturbation theory of Matrices. Detailed illustrations, stimulating examples and exercises as well as a clear explanation of the underlying theory makes this workbook ideal for senior undergraduate and beginning graduate students in applied mathematics as well as science and engineering fields.