In this thesis I discuss some results on the noncommutative (spin) geometry of quantum principal G-bundles. The first part of the thesis is devoted to the study of spectral triples over toral bundles; extending some recent results by L. Dabrowski and A. Sitarz, we introduce the notion of projectable spectral triple for T^n-bundles. Moreover, we work out twisted Dirac operators. We discuss, in particular, the application of these results to noncommutative tori. In the second part of the thesis, instead, we work out a method for constructing real spectral triples over cleft quantum principal G-bundles and we study the properties of these triples and their behaviour under gauge transformations.
Some of the results discussed in this thesis can also be found in the following papers:
arXiv:1305.6185
arXiv:1308.4738
