The proposed project aims at carrying out research on various aspects of quantum gravity, related areas of mathematical physics, certain topics in cosmology, astroparticle physics and physics of black holes. Whereas a satisfactory theory of quantum gravity still remains elusive, some of the assessed properties revealing a nonlocal character of such a theory (i. e. noncommutativity of spacetime coordinates, UV/IR mixing and the holographic principle) has already changed our perspective upon reconciliation of the quantum field theory (QFT) framework with general relativity. A particular attention will be paid to construction, classification and unification of NC spaces. In this regard, a procedure can be defined by making use of rigorous quantum deformation techniques as well as some well known differential algebra concepts and also the Hopf algebra methods. This permits formulation and study of the twist-deformed Riemannian geometry, the twisted version of Einstein's equations and the algebraic structure on kappa-deformed Minkowski space. A further extension of these ideas opens a possibility to study NC effect in black holes. Furthermore, using the tools of QFT in curved spacetime, several aspects of modern cosmology will be covered including those based on the most successful realization of the holographic principle - the AdS/CFT duality. Finally, a class of quantum gravity models will be tested by probing Lorentz violation in observations of rapidly-varying astrophysical sources of high-energy gamma-rays. This requires a full-fledged field-theoretical model with a trouble-free and meaningful quantum loop results, with all UV and IR pathologies being under control.