  A duality theory of bundles of C$^*$-algebras whose fibres are twisted
transformation group algebras is established. Classical T-duality is obtained
as a special case, where all fibres are commutative tori, i.e. untwisted group
algebras for $\Z^n$. Our theory also includes the bundles considered by Mathai
and Rosenberg in their work on non-commutative T-duals, in which they allow
twisted group algebras on one side of the duality.
