Using raising operators and geometric arguments, we establish formulas for the K-theory classes of degeneracy loci in classical types. We also find new determinantal and Pfaffian expressions for classical cases considered by Giambelli: the loci where a generic matrix drops rank, and where a generic symmetric or skew-symmetric matrix drops rank.In an appendix, we construct a K-theoretic Euler class for evenrank vector bundles with quadratic form, refining the Chow-theoretic class introduced by Edidin and Graham. We also establish a relation between top Chern classes of maximal isotropic subbundles, which is used in proving the type D degeneracy locus formulas.Here p and q are the ranks of E and F , respectively, and c(F − E) is the K-theoretic Chern class. The leading term, where m = 0, is precisely Giambelli's determinantal formula in cohomology. 1 We will prove similar formulas for vexillary degeneracy loci, incorporating the stability properties of the cohomology formulas. 2 Our results generalize the formulas of [HIMN] for types A, B, and C. In general, the loci are described by conditions of the form dim(E p i ∩ F q i ) ≥ k i , and the results of [HIMN] are recovered as the special case where k i = i and p i = p for all i.