This book is intended for a graduate course on asymptotic analysis in applied mathematics. The reader is assumed to have a background in differential equations, advanced calculus, complex analysis and matrix algebra at the level of introductory undergraduate courses. The text is self-contained and the exposition is well-written. The book contains chapters about various methods of asymptotic analysis of exponential integrals, ordinary differential equations, ordinary differential equations with parameters and partial differential equations like propagation of waves. The text stresses rigorous analysis in addition to formal manipulation. Asymptotic expansions are rigorously justified and the reader is shown how to obtain solid error estimates. The book is accompanied by a considerable number of well-chosen examples and exercises that illustrate the theory.