Cosmic reionization by starlight from early galaxies affected their evolution, thereby impacting reionization itself. Star formation suppression, for example, may explain the observed underabundance of Local Group dwarfs relative to N -body predictions for cold dark matter. Reionization modelling requires simulating volumes large enough [~ (100 Mpc) 3 ] to sample reionization ‘patchiness’, while resolving millions of galaxy sources above ~10 8 M combining gravitational and gas dynamics with radiative transfer. Modelling the Local Group requires initial cosmological density fluctuations pre-selected to form the well-known structures of the Local Universe today. Cosmic Dawn (‘CoDa’) is the first such fully coupled, radiation-hydrodynamics simulation of reionization of the Local Universe. Our new hybrid CPU-GPU code, ramses-cudaton , performs hundreds of radiative transfer and ionization rate-solver timesteps on the GPUs for each hydro-gravity timestep on the CPUs. CoDa simulated (91Mpc) 3 with 4096 3 particles and cells, to redshift 4.23, on ORNL supercomputer Titan, utilizing 8192 cores and 8192 GPUs. Global reionization ended slightly later than observed. However, a simple temporal rescaling which brings the evolution of ionized fraction into agreement with observations also reconciles ionizing flux density, cosmic star formation history, CMB electron scattering optical depth and galaxy UV luminosity function with their observed values. Photoionization heating suppressed the star formation of haloes below ~2 x 10 9 M , decreasing the abundance of faint galaxies around M AB 1600 = [–10, –12]. For most of reionization, star formation was dominated by haloes between 10 10 –10 11 M , so low-mass halo suppression was not reflected by a distinct feature in the global star formation history. Intergalactic filaments display sheathed structures, with hot envelopes surrounding cooler cores, but do not self-shield, unlike regions denser than 100 〈〉.