This project in mathematical physics is concerned with the mathematical understanding of superconductivity and Bose-Einstein condensation. These physical phenomena are the subject of intense research activity both in the experimental and theoretical physics communities and in mathematics. However, despite a lot of effort, many key questions lack a mathematically rigorous answer. The ambition of the present project is to improve this situation. I plan to analyze both the effective models and the underlying microscopic description of superconductivity and Bose-Einstein condensation. The effective models are (systems of) non-linear partial differential equations, and I will apply recently developed mathematical techniques for their analysis. To mention an important specific problem in this part of the project, I am interested in the appearance of regular (Abrikosov) lattices of vortices. For superconductivity, which I will treat in the Ginzburg-Landau model, it is an experimental fact that this happens when an exterior magnetic field comes close to a critical value. For rotating Bose-Einstein condensates, in the Gross-Pitaevskii model, a similar phenomenon occurs for sufficiently large rotations. However, as yet we are unable to derive these lattices directly from the relevant equations. Even more fundamental are the questions about the microscopic models. The aim here is to prove that the desired condensation actually occurs under conditions relevant to experiment, i.e. to prove that the condensation phenomena are correctly described by our fundamental equations of Nature. The microscopic models are systems with a large number of variables and developing the mathematical techniques necessary for the analysis of such systems is an important question in current research in Mathematics.