### Abstract

Hopf cyclic cohomology is a new cohomology theory for Hopf algebras. It was introduced by Connes and Moscovici in 1998 in the course of their study of transverse index theory on foliated manifolds. Its relation to cyclic cohomology is to some extent similar to the relation between de Rham cohomology and group and Lie algebra cohomology, though this was not the original motivation and this analogy by itself can be of no help in defining it! In the first part of this talk I will give a general introduction to Hopf cyclic cohomology, its origins, the progress made in the last few years, and our current understanding of the subject. In the second half I will present new joint work with Atabey Kaygun where a bivariant Hopf cyclic theory is developed.