  In this paper we extend the classical notion of digraphical and graphical
regular representation of a group and we classify, by means of an explicit
description, the finite groups satisfying this generalization. A graph or
digraph is called regular if each vertex has the same valency, or, the same
out-valency and the same in-valency, respectively. An m-(di)graphical regular
representation (respectively, m-GRR and m-DRR, for short) of a group G is a
regular (di)graph whose automorphism group is isomorphic to G and acts
semiregularly on the vertex set with m orbits. When m=1, this definition agrees
with the classical notion of GRR and DRR. Finite groups admitting a 1-DRR were
classified by Babai in 1980, and the analogue classification of finite groups
admitting a 1-GRR was completed by Godsil in 1981. Pivoting on these two
results in this paper we classify finite groups admitting an m-GRR or an m-DRR,
for arbitrary positive integers m. For instance, we prove that every
non-identity finite group admits an m-GRR, for every m>4.
