The contents of this book are based principally on a course given by the author in 1970-71 at the University of Maryland. Since it was not an express aim of the course to develop a full and complete theory, several important aspects of real analytic Eisenstein series are not included in the present text. The main part of the book gives an expository description of the analytic continuation of Eisenstein series concerning SL(2, R). In addition, spectral decomposition of the space of square integrable auto- morphic functions with respect to a general Fuchsian group is outlined, since it constitutes one of the most interesting applications of real analytic Eisenstein series. However, the primary aim of the book is to summarize the characteristic properties of real analytic Eisenstein series; consequently, items such as purely functional analytic arguments are left in rather abbreviated form.