A fundamental invariant associated to every rational function f:X->P^1 on a compact Riemann surface is its monodromy group, that is, the Galois group of the normal closure of the covering given by f. We determine all indecomposable rational functions of sufficiently large degree n with a specified monodromy group that is not A_n or S_n. We shall explain this classification and its applications to problems in number theory, complex analysis, and complex dynamics. Joint work with Michael Zieve.