  Pairings are particular bilinear maps, and as any bilinear maps they factor
through the tensor product as group homomorphisms. Besides, nothing seems to
prevent us to construct pairings on other abelian groups than elliptic curves
or more general abelian varieties. The point of view adopted in this
contribution is based on these two observations. Thus we present an elliptic
curve free study of pairings which is essentially based on tensor products of
abelian groups (or modules). Tensor products of abelian groups are even
explicitly computed under finiteness conditions. We reveal that the existence
of pairings depends on the non-degeneracy of some universal bilinear map,
called the canonical bilinear map. In particular it is shown that the
construction of a pairing on $A\times A$ is always possible whatever a finite
abelian group $A$ is. We also propose some new constructions of pairings, one
of them being based on the notion of group duality which is related to the
concept of non-degeneracy.
