  Almost Moore mixed graphs\/} appear in the context of the degree/diameter
problem as a class of extremal mixed graphs, in the sense that their order is
one unit less than the Moore bound for such graphs. The problem of their
existence has been considered just for diameter $2$. In this paper, we give a
complete characterization of these extremal mixed graphs for diameters 2 and 3.
We also derive some optimal constructions for other diameters.
