This chapter is from the book "Theta Functions on Riemann Surfaces" by John D. Fay, published by Springer-Verlag in 1973. The book covers both new and classical results in the theory of theta functions on Riemann surfaces, a subject that has seen renewed interest in recent years. Key topics include the relationships 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 author acknowledges the support and assistance provided by Prof. Lars V. Ahlfors and Prof. David Mumford, and notes that the research was supported by the National Science Foundation. The book is structured into several sections, each focusing on different aspects of the theory, and includes a notational index and references.This chapter is from the book "Theta Functions on Riemann Surfaces" by John D. Fay, published by Springer-Verlag in 1973. The book covers both new and classical results in the theory of theta functions on Riemann surfaces, a subject that has seen renewed interest in recent years. Key topics include the relationships 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 author acknowledges the support and assistance provided by Prof. Lars V. Ahlfors and Prof. David Mumford, and notes that the research was supported by the National Science Foundation. The book is structured into several sections, each focusing on different aspects of the theory, and includes a notational index and references.