These notes present new as well as classical results from the theory of theta functions on Riemann surfaces, a subject of renewed interest in recent years - [5], [10]. Topics discussed here include: the relations 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. I wish to express sincere thanks to Prof. Lars V. Ahlfors for his constant help and encouragement over many years, and to Prof. David Mumford for generous assistance at several points in this work. Research for these notes was supported by the National Science Foundation.
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