In this thesis, we study the building block for the description of the granular structure of polycrystalline superconductors - the Josephson junction. We investigate the critical current density as a function of applied magnetic field both analytically and computationally, through the lens of time-dependent Ginzburg-Landau theory (TDGL) in 2D. We derive new analytic expressions for the order parameter distribution near interfaces of arbitrary material properties in 2D, validate them using TDGL simulations and use them to extract the effective upper critical field. These results represent a generalization of the famous work from Saint James and de Gennes to arbitrary grain boundary properties. We then extend this framework to include the transport current flowing across the grain boundary, and obtain analytic expressions for the maximum current density that can flow across the grain boundary, providing a generalization of the in-field work in the literature, to high resistivity grain boundaries. We provide a framework to predict the critical current density across the 2D grain boundary over the entire applied magnetic field range, again validated using TDGL simulations. Crucially, our derived expressions consider arbitrary width in detail, but require no additional free parameters, since the derivation formally includes the complexity near interfaces with arbitrary material parameters. We demonstrate how our analytic extension and treatment is necessary for systems with geometries and material parameters which are representative of commercial high-field superconducting materials. Finally, we address how to apply our understanding of a single Josephson junction to 3D polycrystalline materials
