Abstract. Functional decomposition|whether a function f(x) canbe written as a composition of functions g(h(x)) in a nontrivial way|is an important primitive in symbolic computation systems. The problem of univariate polynomial decomposition was shown to have an e cient solution by Kozen and Landau . Dickerson  and von zur Gathen  gave algorithms for certain multivariate cases. Zippel  showed how to decompose rational functions. In this paper, we address the issue of decomposition of algebraic functions. We showthat the problem is related to univariate resultants in algebraic function elds, and in fact can be reformulated as a problem of resultant decomposition. Wecharacterize all decompositions of a given algebraic function up to isomorphism, and give an exponential time algorithm for nding a nontrivial one if it exists. The algorithm involves genus calculations and constructing transcendental generators of elds of genus zero.