The enumeration of singularities of differend kind is a classical subject of enumerative geometry. In this talk we study the following problem: given a holomorphic map of complex analytic manifolds, what are the cohomology classes on the source and the target manifolds dual to various loci of multisingularities of differnt type? This problem was studied for last 30 years in the framework of intersection theory. The most strong result were the formulas of Kleiman and Katz for the cycles of multiple points valid for maps admitting only singularities of corank 1. We present a new formula valid for all proper finite maps and for multisingularities of any kind in terms of the so called residue polynomials. The motivation for this formula came from topology, namely, from cobordism theory --- the field that looks rather far from algebraic geometry and intersection theory, perhaps, this was the reason why this formula were not observed before. This result demonstrates the unity of mathematics: combinig the methods of completely different domains (intersection theory and cobordism theory in this case) one may obtain new results in both of them.
