We discuss topological T-duality and the associated geometric or topological objects in this thesis. Concretely, it consists of three parts. In this first part we prove two versions of geometric twisted K-homology are equivalent and construct the T-duality transformation for geometric twisted K-homology. This gives a dual picture for T-duality transformation of twisted K-groups. In the second part, we show that T-duality isomorphism of twisted K-theory is unique, which gives rise to the conclusion that T-duality isomorphisms through different approach (e.g. algebraic topology, C*-algebra and groupoid) are the same. We also prove that 2-fold composition of T-duality isomorphism is equal to identity, which is given before in other papers but not proved correctly. In the third part, We discuss T-duality for circle actions. We construct the topological T-duality for countable infinite CW-complexes and use this to describe the T-duality for proper circle actions. Moreover, Mathai and Wu's discussion on the same topic is also equivalent to my construction. We also discuss the relations between this approach and C. Daenzer's groupoid approach.
