This is a concise introduction to Fourier series covering history, major themes, theorems, examples and applications. It can be used to learn this subject, and also to supplement, enhance and embellish undergraduate courses on mathematical analysis. The book begins with a brief summary of the rich history of the subject over three centuries. The subject is presented in a way that enables the reader to appreciate how a mathematical theory develops in stages from a practical problem (such as conduction of heat) to an abstract theory dealing with concepts such as sets, functions, infinity and convergence. The abstract theory will provide unforeseen applications in diverse areas. The book begins with a description of the problem that led Fourier to introduce his famous series. The mathematical problems this leads to are discussed rigorously. Examples, exercises and directions for further reading and research are provided, along with a chapter that provides material at a more advanced level suitable for graduate students. The author demonstrates applications of the theory as well as a broad range of problems. Exercises of varying levels if difficulty are scattered throughout the book. These will help readers test their understanding of the material.