The paper is a research announcement of some of the author's recent results concerning degeneracy loci. Let X be a scheme over a field and \(\phi: F\to E\) a morphism of vector bundles over X. For every \(r\geq 0\) the degeneracy locus of rank r associated with \(\phi\) is defined as \(D_ r(\phi)=\{x\in X,\quad rk(\phi (x))\leq r\}.\) In analogy with the general case it is interesting to consider the situation: \(F=E^{\vee}\) and \(\phi\) is symmetric (resp. antisymmetric). Using the classical Schur S- and Q-polynomials, we describe the ideal of all polynomials in the Chern classes of E and F which describe in a universal way all the cycles supported in \(D_ r(\phi)\). As an application we calculate the Chow groups and Chern numbers of determinantal varieties. The ideals that we construct yield also a generalization of the resultant of two polynomials in elimination theory. For a detailed account see ``Enumerative geometry of degeneracy loci'' (to appear in Ann. Sci.Éc. Norm. Supér.) and ``Algebra-geometric applications of Schur S- and Q-polynomials'' (to appear in Sém. Algèbre, Dubreil-Malliavin).