This thesis is a collection of projects focused on the advancement of the applications of distribution theory undertaken during the past years. It contains 11 chapters, with chapter 1 serving as an introduction to provide an overview of the topics addressed in the thesis and other projects that have been completed, but have not been included in this dissertation. Furthermore, a summary of other academic activities such as participation in international conferences will also be given. The last chapter will conclude the thesis with a summary of the results that have been obtained in the individual chapters, and will focus on future extensions of the work presented. The topics included in this thesis fall into two sections. The first part addresses distribution theory applications in finance, linguistics and natural hazards, with a focus on rank-size distribution functions. We investigate the adequacy of various distributions on these relations and develop new multi-sectioned models. We close the section by introducing a software package that offers an accessible implementation of the multi-sectioned distribution model. The second part is focused around advances in signal processing through aspects of distribution theory. We derive novel more efficient and robust approximation methods for common statistical functions, such as the multivariate Rayleigh distribution and the lognormal characteristic function. We furthermore derive a formulation of the general moments of the round-off errors of random variables of arbitrary distribution. Lastly, a software package which contains implementations for multivariate empirical densities is introduced and their performance is compared to previous algorithms.