Summary: A ditroid is a directed version of a matroid. In this paper we investigate the transversal theory of ditroids. Directed versions of the Rado-Hall and Edmonds-Fulkerson theorems are obtained. Our results provide partial answers to two questions raised by L. Qi [Math. Program., Ser. B 42, No. 3, 579-599 (1988; Zbl 0665.90075)].