An algebraic function field of one variable is closely analogous to an algebraic number field, though there are also some number-theoretic methods available for the latter which have no parallel for the former. Instead there are two other sources of function fields: complex function theory and algebraic geometry. A full development of the geometric connexion would go beyond the framework, so apart from occasional geometric indications, this chapter concentrates on the function-theoretic approach. Each algebraic function field in one variable may be represented by a Riemann surface or, from the geometric point of view, as the field of rational functions on a certain algebraic curve. The places of our function field over the constant field correspond to the points of the Riemann surface. Our main objective, a description of the divisor class group, is accomplished by the Abel-Jacobi theorem. This is a result of function theory, though the actual proofs will be algebraic, as far as possible.