  Cyclic cohomology has been recently adapted to the treatment of Hopf symmetry
in noncommutative geometry. The resulting theory of characteristic classes for
Hopf algebras and their actions on algebras allows to expand the range of
applications of cyclic cohomology. It is the goal of the present paper to
illustrate these recent developments, with special emphasis on the application
to transverse index theory, and point towards future directions. In particular,
we highlight the remarkable accord between our framework for cyclic cohomology
of Hopf algebras on one hand and both the algebraic as well as the analytic
theory of quantum groups on the other, manifest in the construction of the
modular square.
