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This introductory graduate text is based on a course the author regularly teaches to students in applied mathematics, engineering sciences, and physics. Each chapter begins with an introductory development of the material and then transitions into more challenging aspects of the topic, or more recent developments arising from current research. The topics include boundary layer theory, multiple scales, WKB, and homogenization, with examples involving ordinary and partial differential equations, difference equations, integral equations, and delay equations. Many of the exercises are derived from problems arising in current research and are drawn from a wide range of application areas. This edition includes a new chapter on the asymptotic expansion of integrals, as well as new material on the foundations of asymptotic expansions and applications related to the evolutionary behavior of nonlinear systems.