We present cosmological hydrodynamical simulations of eight Milky Way-sized haloes that have been previously studied with dark matter only in the Aquarius project. For the first time, we employ the moving-mesh code AREPO in zoom simulations combined with a comprehensive model for galaxy formation physics designed for large0 cosmological simulations. Our simulations form in most of the eight haloes strongly disc-dominated systems with realistic rotation curves, close to exponential surface density profiles, a stellar-mass to halo-mass ratio that matches expectations from abundance matching techniques, and galaxy sizes and ages consistent with expectations from large galaxy surveys in the local Universe. There is no evidence for any dark matter core formation in our simulations, even so they include repeated baryonic outflows by supernova-driven winds and black hole quasar feedback. For one of our haloes, the object studied in the recent `Aquila' code comparison project, we carried out a resolution study with our techniques, covering a dynamic range of 64 in mass resolution. Without any change in our feedback parameters, the final galaxy properties are reassuringly similar, in contrast to other modelling techniques used in the field that are inherently resolution dependent. This success in producing realistic disc galaxies is reached, in the context of our interstellar medium treatment, without resorting to a high density threshold for star formation, a low star formation efficiency, or early stellar feedback, factors deemed crucial for disc formation by other recent numerical studies.