Theta Functions on Riemann Surfaces [Repost]

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], [i0]. 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.