In this thesis we investigate numerically how ionizing photons emitted by stars in galaxies cause the reionization of the Universe, the transition during which most of the gas in the Universe from a mostly neutral, to a highly ionised state it is in today. To this end, we discuss and improve two techniques for the transport of ionising radiation across cosmological volumes, analyse the sources of ionising photons at high redshifts predicted by a semi-analytical galaxy formation model (GALFORM), and combine these to make consistent model of how reionization proceeds. Our improvements to the hybrid characteristics (HC) radiative transport scheme are significant, making the code faster and more accurate, as demonstrated by our contribution to a code comparison paper (Iliev et al., 2009). Our improvements to the SimpleX radiative transport scheme allow for accurate and significantly better numerically converged calculations of the speeds of ionization fronts of cosmological HII regions. This is accomplished by a much more thorough analysis of how to properly model the density field on the unstructured density field in SimpleX. The dependence of the ionizing emissivity of GALFORM galaxies on various parameters of the model is examined. We show that massive stars formed in abundance because of the assumed top-heavy stellar initial mass function during starbursts in the Baugh et al. (2005) model, triggered by galaxy mergers, are the dominant source of ionizing photons. We show that the luminosity functions predicted by this model are in good agreement with the most recent Hubble Space Telescope results at z ≳ 8. The model also demonstrates that most photons are produced in faint galaxies which are not yet seen in the current data. We then combine the sources predicted by GALFORM with the SimpleX RT scheme to model inhomogeneous reionization including the effects of source suppression. We investigate how the morphology of reionization depends on the model for the sources, which may be crucial for future observations of this cosmic epoch.