We examine the epoch of hydrogen reionization using a new numerical method that allows us to self-consistently couple all the relevant physical processes (gas dynamics, dark matter dynamics, self-gravity, star formation/ feedback, radiative transfer, ionization, recombination, heating and cooling) and evolve the system of coupled equations on the same high resolution mesh. We refer to this approach as direct numerical simulation, in contrast to existing approaches which decouple and coarse-grain the radiative transfer and ionization balance calculations relative to the underlying dynamical calculation. Our method is scalable with respect to the number of radiation sources, size of the mesh, and the number of computer processors employed, and is described in Chapter 2 of this thesis. This scalability permits us to simulate cosmological reionization in large cosmological volumes (~100 Mpc) while directly modeling the sources and sinks of ionizing radiation, including radiative feedback effects such as photoevaporation of gas from halos, Jeans smoothing of the IGM, and enhanced recombination due to small scale clumping. With our fiducial simulation, we find that roughly 2 ionizing photons per baryon is needed to highly ionize the intergalactic medium. The complicated events during reionization that lead to this number can be generally described as inside-out, but in reality the narrative depends on the level of ionization of the gas one defines as ionized. We have updated the formula observers often use for estimating the ionized volume filling fraction formula with a [delta]b and [Tau]rec,eff to get from [Omicron](10%) to [Omicron](1%) consistency with our simulation results. This improvement comes from not using the traditional clumping factor, but instead, considering the history and local effects which were neglected in formulating the original expression. And finally, we have a new upper limit for the escape fraction of ~0.6 from our simulation, which takes into account the photons in the energy density field and photons used to ionize H I