AbstractThe clique number of a digraph D is the size of the largest bidirectionally complete subdigraph of D. D is perfect if, for any induced subdigraph H of D, the dichromatic number defined by Neumann‐Lara (The dichromatic number of a digraph, J. Combin. Theory Ser. B 33 (1982), 265–270) equals the clique number . Using the Strong Perfect Graph Theorem (M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas, The strong perfect graph theorem, Ann. Math. 164 (2006), 51–229) we give a characterization of perfect digraphs by a set of forbidden induced subdigraphs. Modifying a recent proof of Bang‐Jensen et al. (Finding an induced subdivision of a digraph, Theoret. Comput. Sci. 443 (2012), 10–24) we show that the recognition of perfect digraphs is co‐‐complete. It turns out that perfect digraphs are exactly the complements of clique‐acyclic superorientations of perfect graphs. Thus, we obtain as a corollary that complements of perfect digraphs have a kernel, using a result of Boros and Gurvich (Perfect graphs are kernel solvable, Discrete Math. 159 (1996), 35–55). Finally, we prove that it is ‐complete to decide whether a perfect digraph has a kernel.