This Ph.D. dissertation is mainly related to the metaphysics of time and the philosophy of science. It aims to satisfy three main desiderata: (i) to provide an accurate characterization of the intuitive asymmetry between the ‘open future' and the ‘fixed past', (ii) to determine which model of the temporal structure of the world is most appropriate to accommodate the asymmetry, and (iii) to reconcile the asymmetry with our best physics. It argues, in particular, that an ontological characterization of the asymmetry, which can be accommodated by a specific version of the ‘growing block theory' of time, finds some naturalistic support in the causal set approach to quantum gravity.