First published in 1980, this monograph covers a range of basic issues in the classical subject of compact Riemann surfaces. The material was selected according to the author's fancy, but he identifies the theory of theta functions as a primary theme. Topics include conics, cubics, theta functions, the Jacobian variety, quartics and quintics, and the Schottky Relation. Although the work has not been revised, an addendum listing references to the subsequent literature has been provided.