Early versions of parts of this work date back to the mid-sixties, when the third author started to write a book on surfaces. But for several reasons, in particular the appearance of Safarevi6's book, he postponed the project. It
was revived about ten years later, when all three authors were in Leiden.
It is impossible to cover in one book the vast and rapidly developing theory
of surfaces. Choices have to be made, with respect to content as well as to
presentation. We have chosen for a complex-analytic point of view; this distin-
distinguishes our text from most of the existing treatments. Relations with the case
of characteristic p are not discussed.
We hope to have succeeded in writing a readable book; a book that can
be used by non-specialists. The specialist will find very little that is new to
him anyhow.