  We extend topological T-duality to the case of general circle bundles. In
this setting we prove existence and uniqueness of T-duals. We then show that
T-dual spaces have isomorphic twisted cohomology, twisted $K$-theory and
Courant algebroids. A novel feature is that we must consider two kinds of
twists in de Rham cohomology and $K$-theory, namely by degree 3 integral
classes and a less familiar kind of twist using real line bundles. We give some
examples of T-dual non-oriented circle bundles and calculate their twisted
$K$-theory.
