The discovery of topological quantum states of matter has required physicists to look beyond Landau’s theory of symmetry-breaking, previously the main paradigm for<br>studying states of matter. This has led also to the development of new topological theories for describing the novel properties. In this dissertation an investigation in this<br>frontier research area is presented, which looks at the interplay between the quantum geometry of these states, defects and disorder. After a brief introduction to the topological quantum states of matter considered herein, some aspects of my work in this area are described. First, the disorder-induced band structure engineering of topological insulator surface states is considered, which is possible due to their resilience from Anderson localization, and believed to be a consequence of their topological origin.<br>Next, the idiosyncratic behavior of these same surface states is considered, as observed in experiments on thin film topological insulators, in response to competition between<br>hybridization effects and an in-plane magnetic field. Then moving in a very different direction, the uncovering of topological ‘gravitational’ response is explained: the<br>topologically-protected charge response of two dimensional gapped electronic topological states to a special kind of 0-dimensional boundary – a disclination – that encodes spatial curvature. Finally, an intriguing relation between the gravitational response of quantum Hall states, and their response to an apparently unrelated perturbation – nonuniform electric fields is reported. <br>