This seminar reviews some of the global algebraic and geometric structure present in T-duality. Bouwknegt, Evslin and Mathai have given a geometric procedure for handling T-duality for certain non-trivial principal torus bundles. Subsequent work by Mathai and Rosenberg showed that sometimes when there is no geometric T-dual there may be a non-commutative torus bundle playing the same role. This exploited a very similar duality for C-star-algebras known for a couple of decades. This talk will review those developments and their recent extension in collaboration with Bouwknegt and Mathai to more general situations in which the algebraic structure becomes non-associative.