There are several possible definitions of an injective homomorphism of a digraph G to a digraph H . Each of them leads to a colouring parameter for oriented graphs by defining the injective oriented chromatic number of an oriented graph G to be the smallest number of vertices in an oriented graph H for which there is an injective homomorphism of G to H . One possible choice leads to colourings (proper or not) that are injective on both in-neighbourhoods and out-neighbourhoods separately. We will consider these possibilities and the associated colouring parameters that arise from them in terms of complexity, obstructions, critical graphs, cliques, products and bounds. This is joint work with Gary MacGillivray, University of Victoria.