Elementary theories of bilinear mappings are studied. The interest in these questions is explained by the fact that different model-theoretic problems of algebra reduce largely to the corresponding problems for bilinear mappings. Bilinear mappings in rings arise most naturally --- the multiplication operation itself generates them. In groups, the role of a bilinear mapping is played by the commutation operator, pertaining to the corresponding factors of some central series, or, from another point of view, the commutation operator in an adjoint Lie ring. The results obtained are used substantially in classifying finitely generated nilpotent groups according to elementary properties, in describing \(\omega\)-stable, of finite Morley rank, uncountable-categorical rings of characteristic zero (and also the corresponding classes of nilpotent groups). Particular attention in the paper is paid to the formal definability of certain important objects and relations in bilinear mappings. As a consequence, a description is obtained of abstract isomorphisms of bilinear mappings.