We introduce a new suite of radiation-hydrodynamical simulations of galaxy formation and reionization called Aurora. The Aurora simulations make use of a spatially adaptive radiative transfer technique that lets us accurately capture the small-scale structure in the gas at the resolution of the hydrodynamics, in cosmological volumes. In addition to ionizing radiation, Aurora includes galactic winds driven by star formation and the enrichment of the universe with metals synthesized in the stars. Our reference simulation uses 2 × 5123 dark matter and gas particles in a box of size 25 h-1 comoving Mpc with a force softening scale of at most 0.28 h-1 kpc. It is accompanied by simulations in larger and smaller boxes and at higher and lower resolution, employing up to 2 × 10243 particles, to investigate numerical convergence. All simulations are calibrated to yield simulated star formation rate functions in close agreement with observational constraints at redshift z = 7 and to achieve reionization at z ≈ 8.3, which is consistent with the observed optical depth to reionization. We focus on the design and calibration of the simulations and present some first results. The median stellar metallicities of low-mass galaxies at z = 6 are consistent with the metallicities of dwarf galaxies in the Local Group, which are believed to have formed most of their stars at high redshifts. After reionization, the mean photoionization rate decreases systematically with increasing resolution. This coincides with a systematic increase in the abundance of neutral hydrogen absorbers in the intergalactic medium.