Non-centrosymmetric crystals -- especially polar materials -- are capable of producing electric current in response to uniform illumination. This is called the bulk photovoltaic effect (BPVE), which we show can be identified with ``shift current'' theory. Shift currents exhibit unique physics, which are discussed and clarified. A discrete form of the expression required for numerical implementation is derived that allows for robust and efficient calculation from first-principles calculations. The response for BaTiO3 and BiFeO3 is calculated and found to agree well with experiment, and careful analysis of the computed response reveals how the magnitude depends on structural and chemical properties, providing criteria for the search for and design of materials with large response. Additionally, the unique properties of shift currents allow for pure spin photocurrents in antiferromagnets with appropriate symmetry. We predict that these spin currents can be observed in BiFeO3 and hematite (Fe2O3), and calculate the expected response. Topological insulators are a class of materials that are bulk insulators with metallic surface states that take the form of helical Dirac cones protected by time-reversal symmetry. Here we explore phenomena that occur near or at the transition between the trivial and topological insulating phase. In Bi2Se3, the relationship between the topological gap and material strain is investigated and used to explore the topological phase transition. At the critical strain, there exists a bulk 3D Dirac point that is analogous to the 2D Dirac points in graphene, and may be expected to exhibit similar properties. However, this 3D Dirac point is not robust and can be easily gapped by perturbations. We propose that a 3D Dirac point marking a topological phase transition may be protected by spatial symmetries, and outline the constraints under which symmetry groups may contain materials with such points. Based on first principles calculations, we propose BiO2 in the β-cristobalite structure as a metastable 3D Dirac semimetal.