"Characteristic classes of quadratic bundles"

Abstract: A quadratic bundle is an algebraic vector bundle equipped with a
continuously varying family of non-degenerate quadratic forms on the
fibers.  Such an object should have characteristic classes lying in the
motivic cohomology of its base.  Among these you see both (geometric)
Chern classes and (arithmetic) Stiefel-Whitney classes, as well as various
things obtained by combining them.  I will describe the limited amount of
information we presently have about this picture.