  We introduce the notion of Hopf algebroids, in which neither the total
algebras nor the base algebras are required to be commutative. We give a class
of Hopf algebroids associated to module algebras of the Drinfeld doubles of
Hopf algebras when the $R$-matrices act properly. When this construction is
applied to quantum groups, we get examples of quantum groupoids, which are
semi-classical limits of Poisson groupoids. The example of quantum $sl(2)$ is
worked out in details.
