  Generalized connectivity introduced by Hager (1985) has been studied
extensively in undirected graphs and become an established area in undirected
graph theory. For connectivity problems, directed graphs can be considered as
generalizations of undirected graphs. In this paper, we introduce a natural
extension of generalized $k$-connectivity of undirected graphs to directed
graphs (we call it strong subgraph $k$-connectivity) by replacing connectivity
with strong connectivity. We prove NP-completeness results and the existence of
polynomial algorithms. We show that strong subgraph $k$--connectivity is, in a
sense, harder to compute than generalized $k$-connectivity. However, strong
subgraph $k$-connectivity can be computed in polynomial time for semicomplete
digraphs and symmetric digraphs. We also provide sharp bounds on strong
subgraph $k$-connectivity and pose some open questions.
