Just as an algebraic function field of one variable represents a curve, so one of two variables represents a surface. While a study of such fields would go well, it is relatively simple to classify the valuations arising in this case, and this also illuminates the situation in the one-dimensional case. This classification of valuations is based on a paper by Zariski, in which he uses these results to give a reduction of singularities of algebraic surfaces. In function theory the aim is the Abel-Jacobi theorem describing the devisor class group, with occasional geometrical asides to help understanding. Assuming only an undergraduate course in algebra, plus a little acquaintance with topology and complex function theory, the chapter serves as an introduction to more technical works in algebraic number theory, function theory or algebraic geometry by an exposition of the central themes in the subject.