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We say that a (di)graph $G$ has a perfect $H$-packing if there exists a setof vertex-disjoint copies of $H$ which cover all the vertices in $G$. Theseminal Hajnal--Szemer\'edi theorem characterises the minimum degree thatensures a graph $G$ contains a perfect $K_r$-packing. In this paper we provethe following analogue for directed graphs: Suppose that $T$ is a tournament on$r$ vertices and $G$ is a digraph of sufficiently large order $n$ where $r$divides $n$. If $G$ has minimum in- and outdegree at least $ (1-1/r)n$ then $G$contains a perfect $T$-packing. In the case when $T$ is a cyclic triangle, thisresult verifies a recent conjecture of Czygrinow, Kierstead and Molla (forlarge digraphs). Furthermore, in the case when $T$ is transitive we conjecturethat it suffices for every vertex in $G$ to have sufficiently large indegree oroutdegree. We prove this conjecture for transitive triangles and asymptoticallyfor all $r \geq 3$. Our approach makes use of a result of Keevash and Mycroftconcerning almost perfect matchings in hypergraphs as well as the DirectedGraph Removal lemma.

On directed versions of the Hajnal--Szemer\'edi theorem