Topological behavior has emerged as a new paradigm in contemporary condensed matter physics. To date, only a relatively small subset of the myriad of possible topological phenomena has been accessed experimentally or utilized for technological applications. In this research program, we will consider devices synthesized of several accessible components, which can potentially allow access to new topological phases, as well as promote several important technological applications. The program consists of three main research directions. The first will focus on defects in topological phases as platforms for performing topological quantum computation. We will study the general mathematical theory describing the defects, as well as a more microscopic analysis of zero modes bound to defects. We will study the quantum information processing capabilities that can be achieved by controlling the coupling between the defects, as well as by coupling defects to external systems. We will also construct lattice models of interacting defects, and study the possible quantum phases of matter which they give rise to. A second research direction will deal with the photo-current response of hetero-structures containing topological insulators. We will study the possibility of inducing magnetic or electrostatic superlattices on topological insulator surfaces in order to enhance their photo-current harvesting capabilities. These structures may serve as a route for constructing new photo-voltaic devices. A third direction will deal with systems in which the topological phenomenon is induced dynamically by an external periodic drive. We will study theoretically the feasibility of realizing and detecting topological phenomena in a non-equilibrium setting. Such a setting will require us to consider coupling to a heat bath (e.g., phonons), interactions, and may imply a distinct response to spatial and temporal disorder.