## Details

Physics defined on real manifolds and equipped with locality has achieved many successes theoretically as well as in describing our universe. Nevertheless, from a mathematical point of view, it is not as privileged. This thesis explores the possibility of non-Archimedean and non-local physics by studying a range of discrete and continuous models. We begin by discussing how continuous dimensions with different topologies emerge from a sparse coupling lattice model inspired by a recent cold atom experiment proposal. A field theory with both non-Archimedean and Archimedean dimensions is then studied. The propagator of the theory possesses oscillatory behavior. We work out the renormalization and compare the theory with the quantum Dyson's hierarchical model at the criticality. We then proceed to study two non-local field theories: the non-local non-linear sigma model and the non-local quantum electrodynamics. Non-locality altered the behavior of NLSM profoundly by eliminating the Ricci flow and demanding higher-order covariant corrections in the target space. At the same time, the interplay between non-locality and gauge symmetry generates unique RG flows in the non-local QED and makes the theory more controllable. We conclude by introducing a monodromy defect defined in $O(N)$ symmetric conformal theories, which by definition, supports a non-local CFT on the defect. Throughout the journey, we want to convey the idea that non-Archimedean physics and non-local physics exhibits rich and unique phenomena yet are not disconnected from the more ordinary physics.