The Reionization of the Universe is one of the most important and hottest topics of the present day astrophysical research. The most plausible candidates for the reionization process are star-forming galaxies, which according to the predictions of the majority of the theoretical and semi-analytical models should dominate the HI ionizing background at z>3. We have used deep U-band imaging in the COSMOS, CANDELS GOODS-NORTH, and EGS fields in order to estimate the Lyman continuum escape fraction of 69 star forming galaxies at z~3 down to faint magnitude limits (L=0.2L*). We have measured a stringent upper limit (<2%) for the escape fraction of HI ionizing photons from bright galaxies (L>L*), while for the faint population (L=0.2L*) the limit to the escape fraction is <10%. We have computed their contribution to the observed UV background at z~3 and we have found that it is difficult to keep the Universe ionized at z~3 only with star-forming galaxies. We compare our results with recent achievements on the Lyman continuum escape fraction of high-z galaxies and AGN in the literature and discuss future prospects to shed light on the Dark Ages.