  We consider the well-studied cops and robbers game in the context of oriented
graphs, which has received surprisingly little attention to date. We examine
the relationship between the cop numbers of an oriented graph and its
underlying undirected graph, giving a surprising result that there exists at
least one graph $G$ for which every strongly connected orientation of $G$ has
cop number strictly less than that of $G$. We also refute a conjecture on the
structure of cop-win digraphs, study orientations of outerplanar graphs, and
study the cop number of line digraphs. Finally, we consider some the aspects of
optimal play, in particular the capture time of cop-win digraphs and properties
of the relative positions of the cop(s) and robber.
