  Let $T$ be a digraph with vertices $u_1, \dots, u_t$ ($t\ge 2$) and let $H_1,
\dots, H_t$ be digraphs such that $H_i$ has vertices $u_{i,j_i},\ 1\le j_i\le
n_i.$ Then the composition $Q=T[H_1, \dots, H_t]$ is a digraph with vertex set
$\{u_{i,j_i}\colon\, 1\le i\le t, 1\le j_i\le n_i\}$ and arc set
$$A(Q)=\cup^t_{i=1}A(H_i)\cup \{u_{ij_i}u_{pq_p}\colon\, u_iu_p\in A(T), 1\le
j_i\le n_i, 1\le q_p\le n_p\}.$$ The composition $Q=T[H_1, \dots, H_t]$ is a
semicomplete composition if $T$ is semicomplete, i.e. there is at least one arc
between every pair of vertices. Digraph compositions generalize some families
of digraphs, including (extended) semicomplete digraphs, quasi-transitive
digraphs and lexicographic product digraphs. In particular, strong semicomplete
compositions form a significant generalization of strong quasi-transitive
digraphs.
  In this paper, we study the structural properties of semicomplete
compositions and obtain results on connectivity, paths, cycles, strong spanning
subdigraphs and acyclic spanning subgraphs. Our results show that this class of
digraphs shares some nice properties of quasi-transitive digraphs.
