A linking is understood as a pair (G, ~b) such that G is a finite abelian group and ~b is a nonsingular, symmetric bilinear pairing G x G~Q/Z. It is convenient to identify a linking (G, ~) with a matrix which represents ~b relative to the generators of a cyclic splitting of G unless confusion might occur. The linking occurs often in the study of topology. For example, given a closed oriented 3-manifold M, we have a unique linking (~HI(M),q~M) defined by the Poincar~ duality, where "rHI(M ) is the torsion part of the integral homology group Hi(M). The purpose of this paper is to determine completely the structure of the abelian semigroup 9l of all linkings (up to isomorphism) under block sum and to observe that any linking is isomorphic to the linking ~b~ of a closed connected oriented 3-manifold M. To do this, we shall present a complete system of invariants of isomorphic linkings, which arises naturally from our purpose. Such a complete system had already been known by Seifert [ 11] in the case of odd-primary groups, and in general by Burger [2, Satz 5] in terms of Minkowski's beautiful theory. Though they are related directly or indirectly to each other (cf. Fox [5]), we do not discuss here any relation between them.