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The goal of the paper is two-fold. At first, we attempt to give a survey ofsome recent applications of symmetric polynomials and divided differences tointersection theory. We discuss: polynomials universally supported ondegeneracy loci; some explicit formulas for the Chern and Segre classes ofSchur bundles with applications to enumerative geometry; flag degeneracy loci;fundamental classes, diagonals and Gysin maps; intersection rings of G/P andformulas for isotropic degeneracy loci; numerically positive polynomials forample vector bundles.  Apart of surveyed results, the paper contains also some new results as wellas some new proofs of earlier ones: how to compute the fundamental class of asubvariety from the class of the diagonal of the ambient space; how to computethe class of the relative diagonal using Gysin maps; a new formula for pushingforward Schur's Q- polynomials in Grassmannian bundles; a new formula for thetotal Chern class of a Schur bundle; another proof of Schubert's andGiambelli's enumeration of complete quadrics; an operator proof of theJacobi-Trudi formula; a Schur complex proof of the Giambelli-Thom-Porteousformula.

Symmetric polynomials and divided differences in formulas of intersection theory