One of the key questions in observational cosmology is the identification of the sources responsible for cosmic reionization and for keeping the IGM ionized at all times. The general consensus is that a population of faint low-mass galaxies must be responsible for the bulk of the ionizing photons. However, attempts at identifying the ionizing Lyman continuum radiation (LyC) leaking from the individual galaxies have so far been largely unsuccessful at any redshift. In addition to all that, it is not possible to directly study LyC at the epoch of reionization or even in the aftermath (e.g., at any redshift z>4.5), due to the very high cosmic opacity. Thus, the galaxy contribution to the cosmic ionization budget must be inferred from other properties that correlate with their ability to contribute to ionizing radiation, if such properties exist. I will discuss advances on this topic, in particular: (1) the search of sources within the first billion years (z~6-7.5) with deep spectroscopy, (2) the Ly-alpha emission demography at z<7.5 as a signature of the reionization era and (3) the ongoing investigation of ionizing emission at moderate redshift (z~3). I will also present first results from faint luminosity regimes (L1500<0.1L*) at z>3, accessible through strong lensing magnification.