Over any associative ring R it is standard to derive HomR(􀀀;􀀀) using projective resolutions in the rst variable, or injective resolutions in the second variable, and doing this, one obtains Extn R(􀀀;􀀀) in both cases. We examine the situation where projective and injective modules are replaced by Gorenstein projective and Gorenstein injective ones, respectively. Furthermore, we derive the tensor product 􀀀 R 􀀀 using Gorenstein at modules.