The author of this book is one of the leading world experts on multiscale methods, a hot area in applied mathematics. The book is meant to be an introduction, aimed primarily towards graduate students. Part I of the book and Part III of the book are necessarily terse and present the wide range of applications of the ideas, and illustrate their unity. The presentation of the material here is particularly suited to the pedagogical goal of communicating the subject area to the wide range of mathematicians, scientists and engineers who are currently engaged in the use of these tools to tackle the enormous range of applications that require them. Extensions and generalizations of the results presented in these notes, as well as references to the literature, are given in the Discussion and Bibliography section, at the end of each chapter. With the exception of Chapter 1, all chapters are supplemented with exercises.