This is a review of the most important results about the automorphisms of algebraic function fields. It starts with the classical results of Schwarz and Hurwitz for the case of Riemann surfaces and those of Schmid and Roquette for algebraic function fields over an algebraically closed field \(K\) of characteristic \(p\geq 0\). The rest of the paper describes a more recent result of R. Brandt [Über die Automorphismengruppen von algebraischen Funktionenkörpern, Dissertation Univ. Essen (1988; Zbl 0716.11058)], which gives explicit equations for Kummer extensions \(F\) of type \([G_0| q,p]\), where \(p,q\) are distinct primes. (Here \(p\) is the characteristic of the field, \(q\) is the order of a central subgroup \(Z\) of a finite group \(G\) of automorphisms of \(F\) over \(K\) such that the fixed field of \(Z\) is rational and \(G_0\cong G/Z\)).