  A directed graph is set-homogeneous if, whenever U and V are isomorphic
finite subdigraphs, there is an automorphism g of the digraph with U^g=V. Here,
extending work of Lachlan on finite homogeneous digraphs, we classify finite
set-homogeneous digraphs, where we allow some pairs of vertices to have arcs in
both directions. Under the assumption that such pairs of vertices are not
allowed, we obtain initial results on countably infinite set-homogeneous
digraphs, classifying those which are not 2-homogeneous.
