The k edge-disjoint paths problem in digraphs with bounded independence number In 1980, Fortune, Hopcroft, and Wyllie showed that the following algorithmic problem (k-EDP) is NP-complete with k=2: k Edge-Disjoint Paths (k-EDP) Instance: A digraph G, and k pairs (s_1,t_1),...,(s_k,t_k) of vertices of G. Question: Do there exist directed paths P_1,...,P_k of G, mutually edge-disjoint, such that P_i is from s_i to t_i for i=1,...,k? In this talk we will present a polynomial time algorithm to solve k-EDP for fixed k in digraphs with independence number at most 2. We will also talk about progress made towards solving k-EDP in digraphs with independence number at most alpha, for fixed alpha. This is joint work with Paul Seymour.