The neighborhood complex of a graph was introduced by Lovรกsz to provide topological lower bounds on chromatic number. More general homomorphism complexes of graphs were further studied by Babson and Kozlov. Such 'Hom complexes' are also related to mixings of graph colorings and other reconfiguration problems, as well as a notion of discrete homotopy for graphs. Here we initiate the detailed study of Hom complexes for directed graphs. For any pair of directed graphs ๐บ and ๐ป we consider the polyhedral complex โ โโ โ Hom(๐บ, ๐ป) that parametrizes the directed graph homomorphisms ๐ : ๐บ โ ๐ป. This construction can be seen as a special case of the poset structure on the set of multihomomorphisms in more general categories, as introduced by Kozlov, Matsushita, and others. Hom complexes of directed graphs have applications in the study of chains in graded posets and cellular resolutions of monomial ideals.We study examples of directed โ โโ โ Hom complexes and relate their topological properties to certain graph operations including products, adjunctions, and foldings. We introduce a notion of a neighborhood complex for a directed graph and prove that its homotopy type is recovered as the โ โโ โ Hom complex of homomorphisms from a directed edge. We establish a number of results regarding the topology of directed neighborhood complexes, including the dependence on directed bipartite subgraphs, a directed graph version of the Mycielksi construction, as well as vanishing theorems for higher homology. The โ โโ โ Hom complexes of directed graphs provide a natural framework for reconfiguration of homomorphisms of directed graphs. Inspired by notions of directed graph colorings we study the connectivity of โ โโ โ Hom(๐บ, ๐ ๐ ) for ๐ ๐ a tournament, obtaining a complete answer for the case of transitive ๐ ๐ . If ๐บ = ๐ ๐ is also a transitive tournament we describe a connection to mixed subdivisions of dilated simplices. Finally we use paths in the internal hom objects of directed graphs to define various notions of homotopy, and discuss connections to the topology of โ โโ โ Hom complexes.