Noncommutative geometry is a relatively new branch of mathematics that grew out of a fusion of operator algebras and differential geometry. Among most important motivations for the first formalism is quantum mechanics. For the second one, it is the general theory of relativity. The concept of noncommutative or quantum spaces provides a new paradigm in bothmathematics and physics. Due to its significantly new perspective, noncommutative geometry requires the development of a number of tools that are motivat ed by and tested on important examples of quantum groups and spaces.The long-term research aim of this project is twofold. Our first objective is to develop the formalism of topological quantum groups and principal bundles and apply it to crucial problems in mathematics, such as the Baum-Connes conjecture, and in mathematical models of physics, such as Yang-Mills or gauge theory. The second objective concerns the construction of cyclic theory that would both serve as a tool in computing K-theoretical invari ants and be adapted to the symmetry of given examples. In particular, this should address the dimensional drop occurring in quantum-group examples. The success in achieving the aforementioned goals hinges greatly on our ability to import knowledge and skil ls in areas such as spectral triples, graph C*-algebras and factors, Poisson geometry, foliations, multiplier Hopf algebras, Yetter-Drinfeld modules, q-special functions, KK-theory and the Baum-Connes conjecture. To this end, it is necessary to establish n ew collaboration links with experts in these areas of mathematics. This is what the transfer of knowledge programme is aimed to achieve.