Boundary Physics and Bulk Boundary Correspondence in Topological Phases of Matter This thesis extends our understanding of systems of independent electrons by developing a generalization of Bloch’s Theorem which is applicable whenever translational symmetry is broken solely due to arbitrary boundary conditions. The thesis begins with a historical overview of topological condensed matter physics placing the work in context before introducing the generalized form of Blochs Theorem. A cornerstone of electronic band structure and transport theory in crystalline matter Blochs Theorem is generalized via a reformulation of the diagonalization problem in terms of corner modified block Toeplitz matrices and physically by allowing the crystal momentum to take complex values. This formulation provides exact expressions for all the energy eigenvalues and eigenstates of the single particle Hamiltonian. By precisely capturing the interplay between bulk and boundary properties this affords an exact analysis of several prototypical models relevant to symmetry protected topological phases of matter including a characterization of zero energy localized boundary excitations in both topological insulators and superconductors. Notably in combination with suitable matrix factorization techniques the generalized Bloch Hamiltonian is also shown to provide a natural starting point for a unified derivation of bulk boundary correspondence for all symmetry classes in one dimension Chapter1: Introduction Chapter2: Generalization of Blochs theorem to systems with boundary Chapter3: Investigation of topological boundary states via generalized Bloch theorem Chapter4: Matrix factorization approach to bulk boundary correspondence Chapter5: Mathematical foundations to the generalized Bloch theorem Chapter6: Summary and Outlook