A GAP package for braid orbit computation and applications Let G be a finite group. By Riemann’s existence theorem, braid orbits of generating systems for G with product 1 correspond to irreducible familes of coverings of the Riemann sphere with monodromy group G. In this article, the authors describe a GAP package for computing the braid orbits. They then apply these techniques to several interesting cases – in particular to the problem of trying to classify indecomposable rational functions (in characteristic zero) of degree n whose monodromy group is not the alternating or symmetric group of degree n. The group theoretic possibilities have almost been completely worked out by various authors and this program should be able to describe the various possibilities up to equivalence of covers. They also consider the case of indecomposable covers of degree n from the generic Riemann surface of genus g when g=2 or 3 (if g≥4, by results of the reviewer and various authors – see the references– only symmetric and alternating groups of degree n can occur).