HOPF-CYCLIC COHOMOLOGY: categorical approach to the Hochschild and cyclic cohomology (pre-cosimplicial and pre-cocyclic objects in Abelian categories, cyclic cohomology as a derived functor), modular pairs in involution, definition of Hopf-cyclic cohomology, Hopf actions and the characteristic map from the Hopf-cyclic to the usual cyclic cohomology, examples (e.g., universal enveloping Hopf algebra of a Lie algebra)