Zirconium alloys are an important material in the nuclear industry, used as a fuel cladding material in water cooled reactors. However, aqueous corrosion of the alloy results in the production of hydrogen at the interface of the cladding with the water. Some of the hydrogen produced may diffuse through the passive oxide layer and be absorbed by the zirconium in solid solution. It has a deleterious effect on the mechanical properties of the cladding alloy, particularly if enough hydrogen is absorbed to cause precipitation of zirconium hydrides. There are multiple different hydride phases that may form, which have complex interactions with alloying additions, hydrogen concentration, stress and temperature. Of particular note is the phenomenon of Delayed Hydride Cracking (DHC), where the interaction of stress fields around crack-tips is believed to bring about the precipitation of hydrides, which promotes further growth of cracks. Overall, the embrittlement and cracking of the cladding by hydride related mechanisms represents a risk for reactor operators, which can be mitigated by a better understanding of the processes at work. This thesis presents an investigation into the mechanisms of hydride formation in zirconium alloys. In order to understand the mechanisms involved, a perspective of the system is has been sought using the ab-initio, quantum mechanical, atomistic simulation technique of Density Functional Theory (DFT). The first component of this study was an examination of the stability of different alloying additions in zirconium. The alloying additions of chromium, iron, nickel, niobium, tin, vanadium and yttrium have been examined, comparing the energy of them existing in a solid solution or in different inter-metallic structures. These intermetallic structures included multiple Laves phase structures, as well as a variety of other configurations. It was found that the thermodynamic driving forces in this system can be correlated with trends in atomic radii and the relative electronegativities of the different species. These same parameters also correlate with the formation energy differences between the different Laves phase polymorphs. Fe and Cr were found to prefer interstitial sites over substitutional locations in the Zr lattice. The Fe atoms had a similar energy of solution in both tetrahedral and octahedral sites, which may have implications for diffusion pathways. Formation energies of Fe, Ni and Sn based intermetallic compounds were found to be negative, and the Zr2Fe and Zr2Ni intermetallics were metastable. Most elements displayed negative energies of solution in beta-zirconium but positive energies in the alpha-phase, with the exception of Sn (which was negative for both) and Y (which was positive for both). Solutions formed from intermetallics showed a similar trend. Incorporation energies onto vacant sites in the Zr lattice were also investigated. It was found that all of the elements investigated showed a driving force to incorporate onto vacancies in alpha-Zr but some would not incorporate into beta-Zr. Different hydride structures were investigated, including the zeta, gamma, delta and epsilon-phases, and some speculative hydride structures. These were also compared with a large number of structures, which were generated with a random positioning of the H on appropriate sites in the Zr lattice. In alpha-Zr, inserting a H atom on the tetrahedral site had a more negative solution enthalpy than the octahedral site. This was also found to be true when the Zr was in a FCC lattice structure, and may have been related to the relative size of the sites. The gamma and epsilon-hydrides appear to be thermodynamically stable, while the delta hydride was very close in energy, but not quite favourable and thus, not quite stable. The zeta-phase, and speculative phases containing H atoms on octahedral sites were significantly less stable. The precipitation reaction was looked at in the context of variations in temperature and pressure. This was done by using phonon density of states data to calculated thermodynamic properties such as heat capacities and vibrational entropies. It was found that at any temperature, the concentration of the initial solid solution had to be at least 300 ppm before precipitation could become favourable. This implies local concentration of hydrogen atoms within a lattice must be significantly greater than the global concentration normally found in experimental results. Increases in temperature were found to drive the reaction towards solution, as would be expected from experimental results. The entropic components of the overall free energy are the main causes of this effect. Finally, the impact of pressure on hydride precipitation was examined, by applying a range of hydrostatic stresses to the alpha-zirconium lattice. Importantly, it was found that a tensile hydrostatic Zr lattice has a more favourable hydrogen solution enthalpy than a relaxed or compressed lattice. This implies that hydrogen may prefer to diffuse towards areas of the lattice which are under tension, in agreement with the Diffusion First Model of DHC. Different hydrides were placed under compressive and tensile stress, which demonstrated that the normally anisotropic stiffness of zirconium hydride lattice becomes isotropic for a stoichiometric ratio of around ZrH1.66.