In this paper, we give some necessary and sufficient conditions for decomposing the complete bipartite digraphs $${\mathcal D}K_{m, n}$$DKm,n and complete digraphs $${\mathcal D}K_n$$DKn into directed paths $${\mathop {P}\limits ^{\rightarrow }}_{k+1}$$P?k+1 and directed cycles $${\mathop {C}\limits ^{\rightarrow }}_{k}$$C?k with $$k$$k arcs each. In particular, we prove that: (1) For any nonnegative integers $$p$$p and $$q$$q; and any positive integers $$m$$m, $$n$$n, and $$k$$k with $$m\ge k$$m?k and $$n\ge k$$n?k; a decomposition of $${\mathcal D} K_{m, n}$$DKm,n into $$p$$p copies of $${\mathop {P}\limits ^{\rightarrow }}_{k+1}$$P?k+1 and $$q$$q copies of $${\mathop {C}\limits ^{\rightarrow }}_{k}$$C?k exists if and only if $$k(p+q)=2mn$$k(p+q)=2mn, $$p\ne 1$$p?1, $$(m, n, k, p)\ne (2, 2, 2, 3)$$(m,n,k,p)?(2,2,2,3), and $$k$$k is even when $$q>0$$q>0. (2) For any nonnegative integers $$p$$p and $$q$$q and any positive integers $$n$$n and $$k$$k with $$k$$k even and $$n\ge 2k$$n?2k, a decomposition of $${\mathcal D} K_{n}$$DKn into $$p$$p copies of $${\mathop {P}\limits ^{\rightarrow }}_{k+1}$$P?k+1 and $$q$$q copies of $${\mathop {C}\limits ^{\rightarrow }}_{k}$$C?k exists if and only if $$k(p+q)=n(n-1)$$k(p+q)=n(n-1) and $$p\ne 1$$p?1. We also give necessary and sufficient conditions for such decompositions to exist when $$k=2$$k=2 or $$4$$4.