  We study the automorphism groups of countable homogeneous directed graphs
(and some additional homogeneous structures) from the point of view of
topological dynamics. We determine precisely which of these automorphism groups
are amenable (in their natural topologies). For those which are amenable, we
determine whether they are uniquely ergodic, leaving unsettled precisely one
case (the "semi-generic" complete multipartite directed graph). We also
consider the Hrushovski property. For most of our results we use the various
techniques of [3], suitably generalized to a context in which the universal
minimal flow is not necessarily the space of all orders. Negative results
concerning amenability rely on constructions of the type considered in [26]. An
additional class of structures (compositions) may be handled directly on the
basis of very general principles. The starting point in all cases is the
determination of the universal minimal flow for the automorphism group, which
in the context of countable homogeneous directed graphs is given in [10] and
the papers cited therein.
