Summary: We study bubble break-up in homogeneous and isotropic turbulence by direct numerical simulations of the two-phase incompressible Navier-Stokes equations. We create the turbulence by forcing in physical space and introduce the bubble once a statistically stationary state is reached. We perform a large ensemble of simulations to investigate the effect of the Weber number (the ratio of turbulent and surface tension forces) on bubble break-up dynamics and statistics, including the child bubble size distribution, and discuss the numerical requirements to obtain results independent of grid size. We characterize the critical Weber number below which no break-up occurs and the associated Hinze scale \(d_h\). At Weber number close to stable conditions (initial bubble sizes \(d_0\approx d_h)\), we observe binary and tertiary break-ups, leading to bubbles mostly between \(0.5d_h\) and \(d_h\), a signature of a production process local in scale. For large Weber numbers \((d_0> 3d_h)\), we observe the creation of a wide range of bubble radii, with numerous child bubbles between \(0.1d_h\) and \(0.3d_h\), an order of magnitude smaller than the parent bubble. The separation of scales between the parent and child bubble is a signature of a production process non-local in scale. The formation mechanism of these sub-Hinze scale bubbles relates to rapid large deformation and successive break-ups: the first break-up in a sequence leaves highly deformed bubbles which will break again, without recovering a spherical shape and creating an array of much smaller bubbles. We discuss the application of this scenario to the production of sub-Hinze bubbles under breaking waves.