Topological superconductors are a newly discovered class of materials with features uniquely advantageous for quantum computing. They have lately generated an immense amount of attention due to the possibility of them having effective Majorana fermions at surfaces, vortices, and other defects. Approximately one can say that a Majorana fermion is half an electron, or more accurately, in a system with Majorana fermions the wave function of an electron has split up into two separate parts. This non-local property of two Majorana fermions can be used for exceptionally fault-tolerant quantum computing. A quantum computer uses the quantum nature of matter to represent data and preform calculations and can be exponentially faster than any classical supercomputer. However, quantum systems are generally extremely sensitive to disturbances and we are still far from being able to construct useful quantum computers. Topological superconductors with Majorana fermions avoid this extreme sensitivity by using the non-local nature of the Majorana fermions, which automatically make them very robust. The goals of this project are to theoretically 1) discover new and experimentally viable topological superconductors and 2) determine the properties of the Majorana fermions and the conditions necessary for feasible topological quantum computation in real materials. This project was initially focused on several known topological superconductors, such as nanowires proximity coupled to superconductors or chains of magnetic impurities on deposited on top of a superconductor. To avoid multiple known issues in these systems, our focus has during the latter years shifted more to discovering new topological superconductors and also more generically study superconductivity in new and also inhomogeneous materials. Examples of current materials we are studying are various graphene-related materials, such as twisted bilayer graphene, already known to be superconducting, and stacks of rhombohedral graphite with a known topological surface state, as well as topological edge properties of the high-temperature cuprate superconductors. We also study generic properties of superconductors, including aiming to unravel the properties of odd-frequency superconductivity and how inhomogeneity in the form of moiré structures, disorder, and phase crystals change the properties through two consecutive ERC grants. We already have many years of experience studying these types of systems using especially a microscopic lattice tight-binding Bogoliubov-de Gennes (BdG) formalism, which is ideally suited for an accurate description of the superconducting state in topological superconductors. Thanks to medium SNIC grants, we have been able to investigate a large number of systems and we have also incorporated strong correlation effects (at the level of Gutzwiller inhomogeneous mean-field theory) to accurately model the cuprate superconductors. Since a few years ago we also perform ab-initio calculations in order to complement our BdG calculations. Here we use density functional theory (DFT) to both find accurate energy levels and band structures, as well as ab-initio phonon spectra and electron-phonon coupling to access superconductivity. We are planning to continue this research, working towards the overall goals of the project.