In 1974, Kempf and Laksov proved an extension of Giambelli's determinantal formula for the locus of degeneracy of a map of vector bundles. A decade later, Pragacz showed that for maps with symmetries, the appropriate analogue of determinant is a Pfaffian. I will describe recent joint work with William Fulton, in which we identify a new class of degeneracy loci which are represented by Pfaffians, determinants, and combinations of both. These loci correspond to certain signed permutations, which are analogues of vexillary permutations in type A.