  This is the third, and last, of a series of papers dealing with oriented
regular representations. Here we complete the classification of finite groups
that admit an oriented regular representation (or ORR for short), and give a
complete answer to a 1980 question of Laszlo Babai: "Which [finite] groups
admit an oriented graph as a DRR?" It is easy to see and well-understood that
generalised dihedral groups do not admit ORRs. We prove that, with 11 small
exceptions (having orders ranging from 8 to 64), every finite group that is not
generalised dihedral has an ORR.
