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