Universal Schubert polynomials

We introduce polynomials that represent general degeneracy loci for maps of vector bundles. These polynomials specialize to the known classical and quantum forms of single and double Schubert polynomials. This is the final version of the paper, to appear in Duke Math. J. The results are strengthened because Lakshmibai and Magyar have proved a conjecture from the previous version: in the universal case, these loci are Cohen-Macaulay.