The subject of real analytic functions is one of the oldest inmathematical analysis. Today it is encountered early in one'smathematical training： the first taste usually comes rn calculus.While most working mathematicians use real analytic functions fromtime to time in their WOfk， the vast lore of real analyticfunctions remains obscure and buried in the literature. It isremarkable that the most accessible treatment of Puiseux's thcoremis in Lefschetz's quute old Algebraic Geometry， that the clearestdiscussion of resolution of singularities for real analyticmanifolds is in a book review by Michael Atiyah， that there is nocompre hensive discussion in print of the embedding problem forreal analytic manifolds.

　　We have had occasion in our collaborative research to becomeacquainted with both the history and the scope of the theory ofreal analytic functions. It seems both appropriate and timely forus to gather together this information in a single volume. Thematerial presented here is of three kinds. The elementary topics，covered in Chapter 1， are presented in great detail. Even resultslike a real analytic inverse function theorem are difficult to findin the literature， and we take pains here to present such topicscarefully. Topics of middling difficulty， such as separate realanalyticity， Puiseux series， the FBI transform， and related ideas （Chapters 2-4）， are covered thoroughly but rather morebriskly. Finally there are some truly deep and difficult topics：embedding of real analytic manifolds， sub and semi-analytic sets，the structure theorem for real analytic varieties， and resolutionof singularities are disc，ussed and described. But thorough proofsin these areas could not possibly be provided in a volume of modestlength.