| Summary: | <p>Non-locality and contextuality are key features of quantum mechanics that distinguish it from classical physics. We aim to develop a deeper, more structural understanding of these phenomena, underpinned by robust and elegant mathematical theory with a view to providing clarity and new perspectives on conceptual and foundational issues. A general framework for logical non-locality is introduced and used to prove that 'Hardy's paradox' is complete for logical non-locality in all (2,2,<em>l</em>) and (2,<em>k</em>,2) Bell scenarios, a consequence of which is that Bell states are the only entangled two-qubit states that are not logically non-local, and that Hardy non-locality can be witnessed with certainty in a tripartite quantum system. A number of developments of the unified sheaf-theoretic approach to non-locality and contextuality are considered, including the first application of cohomology as a tool for studying the phenomena: we find cohomological witnesses corresponding to many of the classic no-go results, and completely characterise contextuality for large families of Kochen-Specker-like models. A connection with the problem of the existence of perfect matchings in k-uniform hypergraphs is explored, leading to new results on the complexity of deciding contextuality. A refinement of the sheaf-theoretic approach is found that captures partial approximations to locality/non-contextuality and can allow Bell models to be constructed from models of more general kinds which are equivalent in terms of non-locality/contextuality. Progress is made on bringing recent results on the nature of the wavefunction within the scope of the logical and sheaf-theoretic methods. Computational tools are developed for quantifying contextuality and finding generalised Bell inequalities for any measurement scenario which complement the research programme. This also leads to a proof that local ontological models with `negative probabilities' generate the no-signalling polytopes for all Bell scenarios.</p>
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