  Given a graph $G$ with $n$ vertices and an Abelian group $A$ of order $n$, an
$A$-distance antimagic labelling of $G$ is a bijection from $V(G)$ to $A$ such
that the vertices of $G$ have pairwise distinct weights, where the weight of a
vertex is the sum (under the operation of $A$) of the labels assigned to its
neighbours. An {$A$-distance magic labelling} of $G$ is a bijection from $V(G)$
to $A$ such that the weights of all vertices of $G$ are equal to the same
element of $A$. In this paper we study these new labellings under a general
setting with a focus on product graphs. We prove among other things several
general results on group antimagic or magic labellings for Cartesian, direct
and strong products of graphs. As applications we obtain several families of
graphs admitting group distance antimagic or magic labellings with respect to
elementary Abelian groups, cyclic groups or direct products of such groups.
