  Let $R$ be a commutative Noetherian ring. In this paper, we study those
finitely generated $R$-modules whose Cousin complexes provide Gorenstein
injective resolutions. We call such a module a G-Gorenstein module.
Characterizations of G-Gorenstein modules are given and a class of such modules
is determined. It is shown that the class of G-Gorenstein modules strictly
contains the class of Gorenstein modules. Also, we provide a Gorenstein
injective resolution for a balanced big Cohen-Macaulay $R$-module. Finally,
using the notion of a G-Gorenstein module, we obtain characterizations of
Gorenstein and regular local rings.
