Galaxy formation is one of the central problems of Physical Cosmology. Neutral hydrogen plays an important role, linking the collapse of cooling gas into haloes with the formation of stars. Lyman alpha, hydrogen’s strongest spectral line, can directly probe neutral hydrogen in the high redshift Universe. Lyα can be observed in absorption in Damped Lyman Alpha systems (DLAs): high Hi column density regions that dominate the neutral gas content of the Universe between z ∼ 0 − 5. Lyα in emission is an important signature of early, starforming galaxies. Both populations, however, present significant theoretical challenges. As part of my thesis, I have developed a Monte Carlo Lyα radiative transfer code to investigate models of early galaxies. Rauch et al. (2008) performed an ultra-deep spectroscopic survey and discovered a new population of very faint, spatially extended Lyα emitters, which they claimed to be the long-sought host galaxies of DLAs at z ∼ 3. I show here that a simple analytical model, which reproduces the incidence rate and kinematics of DLAs in the context of [Lambda]CDM models for structure formation, also reproduces the size distribution of the faint Lyα emitters for plausible parameters, which supports their identification as DLA host galaxies. The model suggests that galaxies in haloes with vc ~ 100−150 km s−1 account for the majority of DLA host galaxies, and that these galaxies at z ~ 3 are the building blocks of typical present-day galaxies like our Milky Way. I further use my newly developed Lyα code to perform detailed 1D radiative transfer calculations, investigating the spatial and spectral distribution of Lyα emission due to star formation at the centre of DLAs, and its dependence on the spatial and velocity structure of the gas. The modelling reproduces the observed properties of both DLAs and the faint Lyα emitters, including the velocity width and column density distribution of DLAs and the large observed spatial extent of the faint emitters. In the model, haloes hosting DLAs retain up to 20% of the cosmic baryon fraction in the form of neutral hydrogen. The scattering of Lyα photons at the observed radii, which can be as large as 50 kpc, requires the bulk velocity of the gas at the centre of the haloes to be moderate. I furthermore perform 3D Lyα radiative transfer simulations, building on numerical simulations of galaxy formation that include galactic winds and gas infall. The Lyα emission region is shown to be larger and smoother than the cross-section for damped absorption by ~ 50%, with Lyα photons scattered effectively by gas with column densities >~ 1017 cm−2. The spectra typically show two peaks, with the relative strength of the red (blue) peak being a reflection of the relative contribution of outflow (inflow) in the velocity profile. There is considerable variation in the observed line profile and spectral intensity with viewing angle. These more realistic models support many of the simplifying assumptions of my previous models, and have the potential to probe the important role of galactic winds in protogalaxies. The main conclusion is that the faint population of Lyα emitters are indeed the longsought host population of DLAs. Ultra-faint observations of Lyα emission have exceptional potential to directly probe the spatial distribution and kinematics of neutral hydrogen in early galaxies.