We prove a local version of the circular law for non-hermitian random matrices and products of independent non-Hermitian matrices under weak moment conditions. Our results generalise recent results of P. Bourgade, H.-T. Yau, J. Yin and Y. Nemish. We apply Stein's method and some new ideas which help to simplify the proof of the local laws. The talk will be based on joint results with F. Goetze and A. Naumov.