The first direct numerical simulations of turbulent Hall dynamos are presented. The evolution of an initially weak and small-scale magnetic field in a system maintained in a stationary regime of hydrodynamic turbulence (by a stirring force at a macroscopic scale) is studied to explore the conditions for exponential growth of the magnetic energy. The Hall current is shown to have a profound effect on turbulent dynamo action; it can strongly enhance or suppress the generation of the large-scale magnetic energy depending on the relative values of the length scales of the system.