I'll give a historical introduction to the theory of orthogonal Galois theoretic characteristic classes of schemes, starting with Serre's 1984 paper on the Hasse-Witt invariant of the trace form of an etale algebra over a field and Frohlich's generalization, and ending with more recent work by Cassou-Nogues, Erez, Taylor, and T. Saito on geometric generalizations of Serre's original formula. I'll introduce the general concept of symmetric vector bundles on a scheme and their (arithmetic) Stiefel-Whitney classes in etale cohomology, global Hasse-Witt invariants, and generalized Frohlich twists. I'll also sketch the usefulness of these invariants in classifying trace forms of etale algebras, studying embedding problems, finding special kinds of normal bases, and studying the L-functions of "orthogonal motives". Finally, I'll discuss some related open questions.