The subject of real analytic functions is one of the oldest in modern mathematics and is the wellspring of the theory of analysis, both real and complex. To date, there is no comprehensive book on the subject, yet the tools of the theory are widely used by mathematicians today. Key topics in the theory of real analytic functions that are covered in this text and are rather difficult to pry out of the literature include: the real analytic implicit function theorem, resolution of singularities, the FBI transform, semi-analytic sets, Faà di Bruno's formula and its applications, zero sets of real analytic functions, Lojaciewicz's theorem, Puiseaux's theorem. New to this second edition are such topics as:* A more revised and comprehensive treatment of the Faà di Bruno formula * An alternative treatment of the implicit function theorem * Topologies on the space of real analytic functions * The Weierstrass Preparation Theorem This well organized and clearly written advanced textbook introduces students to real analytic functions of one or more real variables in a systematic fashion. The first part focuses on elementary properties and classical topics and the second part is devoted to more difficult topics. Many historical remarks, examples, references and an excellent index should encourage student and researcher alike to further study this valuable and exciting theory.