  We define a version of Hochschild homology and cohomology suitable for a
class of algebras admitting compatible actions of bialgebras, called module
algebras. We show this (co)homology, called Hopf--Hochschild (co)homology, can
also be defined as a derived functor on the category of representations of a
crossed product algebra. We investigate the relationship of our theory with
Hopf cyclic cohomology and also prove Morita invariance of the Hopf--Hochschild
(co)homology.
