We introduce a homotopy theory of digraphs (directed
graphs) and prove its basic properties, including the relations to
the homology theory of digraphs constructed by the authors in
previous papers. In particular, we prove the homotopy invariance of
homologies of digraphs and the relation between the fundamental
group of the digraph and its first homology group.
The category of (undirected) graphs can be identified by a natural
way with a full subcategory of digraphs. Thus we obtain also
consistent homology and homotopy theories for graphs.