  This paper is an introduction to Hopf cyclic cohomology with an emphasis on
its most recent developments. We cover three major areas: the original
definition of Hopf cyclic cohomology by Connes and Moscovici as an outgrowth of
their study of transverse index theory on foliated manifolds, the introduction
of Hopf cyclic cohomology with coefficients by
Hajac-Khalkhali-Rangipour-Sommerhauser, and finally the latest episode on
unifying the coefficients as well as extending the notion to more general
settings beyond Hopf algebras. In particular, the last section discusses the
relative Hopf cyclic theory that arises in the braided monoidal category
settings.
