This text covers Riemann surface theory from elementary aspects to the
fontiers of current research. Open and closed surfaces are treated with
emphasis on the compact case. Basic tools are developed to describe the
analytic, geometric, and algebraic properties of Riemann surfaces and
the Abelian varities associated with these surfaces. For this new
edition, the material has been brought up-to-date, and erros have been
corrected. The book should be of interest no only to pure
mathematicians, but also to physicists interested in string theory and
related topics.