A first-principles scaling law, based on turbulent transport considerations, and a multimachine database of density limit discharges from the ASDEX Upgrade, JET, and TCV tokamaks, show that the increase of the boundary turbulent transport with the plasma collisionality sets the maximum density achievable in tokamaks. This scaling law shows a strong dependence on the heating power, therefore predicting for ITER a significantly larger safety margin than the Greenwald empirical scaling [Greenwald et al., Nucl. Fusion, 28, 2199 (1988)] in case of unintentional high-to-low confinement transition.