These lecture notes present new and classical results from the theory of theta functions on Riemann surfaces, a topic that has regained interest in recent years. The content covers various aspects, including the relationship between theta functions and Abelian differentials, theta functions on degenerate Riemann surfaces, Schottky relations for surfaces of special moduli, and theta functions on finite bordered Riemann surfaces. The work is edited by A. Dold and B. Eckmann, and published by Springer-Verlag. The author, John D. Fay, is affiliated with the University of Maryland, College Park. The notes are supported by the National Science Foundation. The book is divided into six chapters: Riemann's Theta Function, The Prime Form, Degenerate Riemann Surfaces, Cyclic Unramified Coverings, Double Ramified Coverings, and Bordered Riemann Surfaces. It also includes a notational index and a list of references. The book is published with ISBN numbers for both the German and English editions. The work is protected by copyright, and reproduction requires permission from the publisher. The preface acknowledges the support of Prof. Lars V. Ahlfors and Prof. David Mumford, and thanks them for their help and assistance. The book was published in 1973.These lecture notes present new and classical results from the theory of theta functions on Riemann surfaces, a topic that has regained interest in recent years. The content covers various aspects, including the relationship between theta functions and Abelian differentials, theta functions on degenerate Riemann surfaces, Schottky relations for surfaces of special moduli, and theta functions on finite bordered Riemann surfaces. The work is edited by A. Dold and B. Eckmann, and published by Springer-Verlag. The author, John D. Fay, is affiliated with the University of Maryland, College Park. The notes are supported by the National Science Foundation. The book is divided into six chapters: Riemann's Theta Function, The Prime Form, Degenerate Riemann Surfaces, Cyclic Unramified Coverings, Double Ramified Coverings, and Bordered Riemann Surfaces. It also includes a notational index and a list of references. The book is published with ISBN numbers for both the German and English editions. The work is protected by copyright, and reproduction requires permission from the publisher. The preface acknowledges the support of Prof. Lars V. Ahlfors and Prof. David Mumford, and thanks them for their help and assistance. The book was published in 1973.