This PhD thesis is composed of a set of studies utilising Density Functional Theory, within the Periodic Electrostatic Embedded Cluster Method, to computationally study the surface chemistry of the actinide oxides; specifically UO2 and PuO2. Aside from general scientific curiosity, the motivation behind the project stems from the United Kingdom's current plutonium storage strategy. Holding the largest stockpile of civilian plutonium in the world, stored as its fluorite-structured oxide PuO2, it is vital to understand the chemistry that occurs between the PuO2 material and its immediate environment within the storage canisters in order to safely and effectively repurpose it for further use, or dispose of in a more permanent fashion. Chapters 1-3 of this thesis describe in more detail the motivation behind the project, and offer a detailed overview of the theories and software used. Chapter 4 investigates the energies required to introduce a single neutral oxygen vacancy to the {111}, {110} and {100} surfaces and immediate subsurfaces of both UO2 and PuO2. In addition, interactions between single water molecules and the substoichiometric {100} surface are explored finding, in agreement with published work, that 'vacancy-healing' dissociative-type geometries are preferable. Chapter 5 details the first computational study performed regarding interactions between hydrogen chloride (HCl) and stoichiometric {111}, {110} and {100} surfaces of PuO2, with adsorption energies translated to a set of desorption temperatures via molecular thermodynamics. Geometries in which the chlorine atom from HCl 'heals' the vacancy site were found to be the most stable of all considered, with accordingly high desorption temperature requirements. The electronic structure of the substoichiometric surfaces is also explored, by comparing projected density of states and spin density isosurfaces. In Chapter 6, the synergistic effects of co-adsorbing H2O and HCl are considered, as a first step towards modelling a more realistic, wet adsorbing environment for HCl. Stabilising chlorine-acceptor hydrogen bonds have been identified, indicating a strong synergy between the adsorption of HCl and H2O on the PuO2 surface.