This is a leisurely survey of compact Riemann surfaces from the point of view of complex function theory rather than of algebraic geometry; so it focuses on functions and differential forms, uses the universal covering space freely, and assumes the standard properties of topological spaces. The deepest and most subtle parts of the subject really involve functions of several complex variables as well as functions of a single complex variable; so the tools of several complex variables, such as sheaf theory, are used fairly freely. The relevant background material that is assumed for reading this book, such as some standard properties of holomorphic functions of several variables, the theory of sheaves and sheaf cohomology, the topology of surfaces, the cohomology of groups, the structure of complex tori in several dimensions, are included as appendices, so that the book should be accessible to anyone who has a standard undergraduate background and is willing to go to the appendices as necessary. The book is a work in progress; included here are those chapters that are in fairly final form, and more will be added from time to time as time and enthusiasm permit. The eventual form of the book is hinted at by the table of contents, which lists other chapters currently being revised or at least being thought about. The material consists to a large extent of topics covered in graduate courses over a number of years at Princeton; some parts derive from earlier graduate courses and have been published previously in one form or another, but they have been fairly extensively rewritten and expanded for inclusion here. I must thank a great many graduate students who have suffered from these courses over the years; their comments, suggestions, corrections, and inspiration have been of fundamental importance, and I regret that I cannot acknowledge all of them by name here.