In recent years a very good understanding of the kinematical structures provided by candidate theories of quantum gravity, such as Loop Quantum Gravity, Spin Foam Models and Causal Dynamical Triangulations, has been obtained. The next key step is to approach the dynamics of these theories, which is encoded in the physical, that is diffeomorphism invariant, observables of the theory. However, the construction of these quantities is extremely complicated, due to the non-linear structure of the theory. This project aims at developing computational and perturbative techniques in order to construct physical observables in quantum gravity theories, to investigate their properties, and to consider implications in particular for cosmology. Perturbative methods for the computation of gauge invariant observables have become available recently. These methods have to be improved and further developed. This will be achieved by considering and comparing non-perturbative and perturbative approaches to quantum gravity. If successful the project will have an impact on our understanding of classical and quantum dynamics in general relativity. This project will enable the applicant to return to Europe, to build up a network of collaborators, and to work at one of the leading European centres for research in quantum gravity.