On topological Pontryagin classes

The Pontryagin classes of a real vector bundle can be defined via Chern classes of its complexification, and appear in Hirzebruch's formula for the signature of a smooth 4n-dimensional manifold for example.  It was realized long ago that Pontryagin classes can be defined more generally for topological bundles, that is, bundles with fibers homeomorphic to euclidean spaces, even in the absense of linear structures.  I will recall a bit of the classical theory of Pontryagin classes for topological bundles, and discuss some new developments including joint work with Randal-Williams on algebraic independence.