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This thesis covers the application of the local density approximation of density functional theory to a variety of related processes in germanium and silicon. Effort has been made to use calculated results to explain experimentally observed phenomena. The behaviour of vacancies and vacancy clusters in germanium has been studied as these are the dominant intrinsic defects in the material. Particular attention was paid to the annealing mechanisms for the divacancy as a precursor to the growth of the larger clusters, for which the electrical properties and formation energies have been studied. Some preliminary work is also presented on the germanium self-interstitial structure and migration paths. Attention was then turned to a selection of dopant-vacancy defects in both silicon and germanium. An effort was made to explain recent experimental observations in silicon through investigating a number of defects related to the arsenic E-centre. Following this, the properties of donor-vacancy clusters in germanium were studied, and comparison with the results calculated for silicon suggest a significant parallel between the behaviour of the defects and dopants in the two materials. Finally, extensive work was performed on the diffusion of phosphorus and boron in germanium. Diffusion of both dopants was studied via interstitial and vacancy mediated paths as well as by a correlated exchange path not involving any intrinsic defects. The results obtained confirmed current theories of the mechanisms involved in the diffusion of the two defects, while also expanding the knowledge of other paths and giving Fermi level dependences for the energy and mechanism for diffusion of the two defects. Boron diffusion was found to exhibit strong Meyer-Neldel rule effects, which are used to explain the unusually high diffusivity prefactors and energy barriers calculated from experimental measurements for this dopant.