Directed graphs without short cycles In a directed graph G, a feedback arc set is a collection of edges whose deletion makes G acyclic. Extending a result of Chudnovsky, Seymour, and Sullivan, we give an upper bound on the size of the minimum feedback arc set in digraphs without short directed cycles. Our result can be also used to answer a question of Yuster concerning cycles of almost given length in digraphs. This is joint work with Peter Keevash and Benny Sudakov.