I shall explain several situations where Hopf algebras naturally appeared in cyclic homology theory and how this led to Connes-Moscovici's discovery of Hopf cyclic cohomology. I will then explain my work with collaborators on extending the theory through various alternative approaches. A key role in the new approach is played by Yetter-Drinfeld modules, a concept of fundamental importance in quantum group theory and low dimensional topology. I shall end with explaining ongoing work on putting the whole theory on a categorical framework and thus extending it beyond Hopf algebras.