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The work provides answers to the following questions:  How can the dynamics of quantum states of matter and geometry be defined in a diffeomorphism-invariant way? What is the relevant space of physical states and which operators act on it? How are the quantum states related to probabilities in the absence of a preferred time? The answers can provide a further part of the route to constructing a fundamental theory of quantum gravity. 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It also describes the deployment of the method used in ATLAS to correct for the distortions in jet energy measurements caused by additional proton–proton interactions. The thesis provides a detailed introduction to understanding jets and dijet searches at the LHC. The experiments were carried out at two record collider centre-of-mass energies (8 and 13 TeV), probing smaller distances than ever before. Across a broad momentum transfer range, the proton constituents (quarks and gluons) display the same kinematical behaviour, and thus still appear to be point-like. Data are compared to predictions corrected for next-to-leading order quantum chromodynamics (NLO QCD) as well aselectroweak effects, demonstrating excellent agreement. 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To date, NR has mainly been used for simulating binary black hole and neutron star mergers like those detected recently by LIGO. Its use as a tool in fundamental problems of gravity and cosmology is novel, but rapidly gaining interest. In this thesis, the author investigates the initial condition problem in early universe cosmology – whether an inflationary expansion period could have “got going” from initially inhomogeneous conditions – and identifies criteria for predicting the robustness of particular models. 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While their important role in the evolution of the Universe has been well established—namely, that DM serves as the building blocks of galaxies, and that DE accelerates the expansion of the Universe—their true nature remains elusive. In the first half, the authors consider ‘sterile neutrino’ DM, motivated by recent claims that these particles may have finally been detected. Using sophisticated models of galaxy formation, the authors find that future observations of the high redshift Universe and faint dwarf galaxies in the Local Group can placestrong constraints on the sterile neutrino scenario. In the second half, the authors propose and test novel numerical algorithms for simulating Universes with a ‘modified’ theory of gravity, as an alternative explanation to accelerated expansion. 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Nevertheless, long-standing open theoretical issues, as well as possible new observational ones arising from the explosive development of cosmology in the last two decades, offer the motivation and lead a large amount of research to be devoted in constructing various extensions and modifications. In this review all extended theories and scenarios are first examined under the light of theoretical consistency, and are then applied in various geometrical backgrounds, such as the cosmological and the spherical symmetric ones. Their predictions at both the background and perturbation levels, and concerningcosmology at early, intermediate and late times, are then confronted with the huge amount of observational data that astrophysics and cosmology has been able to offer in the last two decades. Theories, scenarios and models that successfully and efficiently pass the above steps are classified as viable and are candidates for the description of Nature, allowing readers to get a clear overview of the state of the art and where the field of modified gravity is likely to go. 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Classical elementary systems are Hamiltonian phase spaces, which are associated with co-adjoint orbits of the relativity group. On the other hand, quantum elementary systems are associated with (projective) unitary irreducible representations of the (possibly extended) relativity group (or one of its covering). This study emphasizes the conceptual issues arising in the formulation of such systems and discusses known results in a mathematically rigorous way. Particular attention is paid to: “smooth” transition from classical to quantum theory; physical content under vanishing curvature, from the point of view of a local(“tangent”) Minkowskian observer; and thermal interpretation (on the quantum level), in the sense of the Gibbons-Hawking temperature. 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