In this thesis we present details of the design, development and application of a large scale exact diagonalisation code named DoQO (Diagonalisation of Quantum Observables). Among the features of this code are its ability to exploit physical symmetries and the fact that it has been designed to run in parallel to take advantage of modern high performance computing resources. The primary motivation for developing this code has been the investigation of exotic phases in quantum lattice models, and in particular of topological phases. A signicant portion of the thesis concerns the investigation of supersymmetric lattice models, which involves signicant use of the developed DoQO code. These are a relatively new (2003) family of models consisting of spinless fermions hopping on a lattice with the interactions tuned to a point where the spectrum exhibits supersymmetry. These models are extremely rich in the physics that they exhibit. Among the phases believed to exist in these models are critical, super-frustrated and super-topological phases. DoQO was also employed to investigate nite size eects in the Kitaev honeycomb lattice model. This is a spin model which exhibits both abelian and non abelian topological phases.