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The programme of 'constructive axiomatics', promulgated by Hans Reichenbach in 1924, seeks to build up the architecture of our best theories of physics from basic axioms supposedly imbued with immediate and indubitable empirical content. Taking inspiration from Reichenbach, Hermann Weyl proposed his own 'causal-inertial' approach to the constructive axiomatization of Einstein's general relativity, according to which a relativistic spacetime can be constructed solely from the trajectories of light rays and freely-falling particles; this project, however, came to fruition only in 1972, with the constructive axiomatization of general relativity due to Ehlers, Pirani, and Schild ('EPS'). One century since Reichenbach, and fifty years since EPS, Constructive Axiomatics for Spacetime Physics is a celebration of the constructive axiomatic methodology. It achieves four main tasks. First, it provides a thoroughgoing presentation of the EPS axiomatization, closing missing loopholes, identifying problematic axioms, and so forth-in this way, one gains a much-improved appreciation of the extent to which a causal-inertial approach to general relativity might succeed, and of what such an approach might offer. Second, it synthesizes and assesses the vast but disparate literature on constructive axiomatics which has arisen over the past century and sets the methodology in its proper philosophical context. Third, it generalizes the approach to apply to quantum spacetimes. And fourth, it applies the approach to the context of non-relativistic spacetime physics. All in all, the book demonstrates that constructive axiomatics is live-and-kicking; the book will become the go-to resource for this way of philosophizing about the nature of space and time.