In topological T-duality one usually considers principal torus bundles. Connections with mirror symmetry suggest that T-duality should be applicable to more general torus fibrations. In this direction we consider topological T-duality for torus bundles with non-trivial monodromy. In particular for general circle bundles we prove existence and uniqueness of the T-duals. We consider also the behavior of twisted cohomology and twisted K-theory under such T-dualities.