A very old problem asks for the degree of a variety defined by rank conditions on matrices. The story of the modern approach begins in the 1970's, when Kempf and Laksov proved that the degeneracy locus for a map of vector bundles is given by a certain determinant in their Chern classes. Since then, many variations have been studied -- for example, when the vector bundles are equipped with a symplectic or quadratic form, the formulas become Pfaffians. I will describe recent extensions of these results -- beyond determinants and Pfaffians, and beyond ordinary cohomology -- including my joint work with W. Fulton, as well as work of several others.