We give an overall picture of our present understanding of the effect of the break-up of stable weakly bound nuclei on their fusion cross section with light, medium and heavy mass nuclei, at energies above the Coulomb barrier. The discussion is based mostly on recent data obtained by our group in collaborative experiments with ANU, USP and Tandar Laboratories. We conclude that there is complete fusion suppression for heavy targets, due to the loss of flux, corresponding to the occurrence of incomplete fusion of one of the break-up fragments. For medium and light mass targets, the incomplete fusion is negligible and therefore is no complete fusion suppression.
