Given any subvariety Y of the type A flag variety GL(n,C)/B which is stable under the left action of the maximal torus T consisting of diagonal matrices, one can ask for a formula for the (ordinary or T-equivariant) cohomology class of Y.  By "a formula" we mean a polynomial in the Chern classes of tautological bundles which generate H_T^*(G/B).  Since H_T^*(G/B) is a polynomial ring modulo relations, there is not just one formula for [Y], but rather an entire coset's worth of representatives, each of which gives a different formula.  This raises the question of what makes one formula preferable to another.  I will recall work of A. Knutson and E. Miller which gives geometric justification for Schubert polynomials as the preferred polynomial representatives of classes of Schubert varieties.  I will then describe the results of my own efforts, in joint work with Alexander Yong, to adapt the viewpoint and techniques of Knutson-Miller to other families of orbit closures on GL(n,\C)/B.