The Giambelli–Thom–Porteous formula expresses the class of a degeneracy locus as a certain determinant in the Chern classes of the vector bundle involved. The conditions defining such loci can be encoded in what Lascoux and Schutzenberger called “vexillary” permutations. I will survey these formulas and their recent extensions to degeneracy loci of other types, where Pfaffians replace determinants, and certain signed permutations play the role of vexillary permutations. The focus will be on the interplay between geometry, algebra, and combinatorics.