  A theory of characteristic classes of vector bundles and smooth manifolds
plays an important role in the theory of smooth manifolds. An investigation of
reasonable notions of characteristic classes of singular spaces started since a
systematic study of singular spaces such as singular algebraic varieties. We
make a quick survey of characteristic classes of singular varieties, mainly
focusing on the functorial aspects of some important ones such as the singular
versions of the Chern class, the Todd class and the Thom--Hirzebruch's L-class.
Then we explain our recent "motivic" characteristic classes, which in a sense
unify these three different theories of characteristic classes. We also discuss
bivariant versions of them and characteristic classes of proalgebraic
varieties, which are related to the motivic measures/integrations. Finally we
explain some recent work on "stringy" versions of these theories, together with
some references for "equivariant" counterparts.
