Exploring the plethora of possibilities provided by solid-state systems to realize exotic many-body phases is not only motivated by fundamental questions but also by potential quantum technological applications. In both cases, it is important to have control over the properties of the system in order to engineer the phase of interest, to have a clear theoretical understanding of the microscopic physics, and to be able to probe it. In this regard, superlattice systems have recently brought many exciting results: e.g., the moire lattice that emerges when two layers of graphene are twisted induces correlated phenomena, akin to high-temperature superconductors. Furthermore, artificially arranged atoms on surfaces have become popular tools to design electronic bands. SuperCorr will explore the vast set of possibilities provided by these tunable systems to engineer novel correlated many-body physics, propose ways to probe it, and advance our understanding of the complex phase diagrams of quantum matter. More specifically, we will address key questions related to several different graphene moire systems, such as the origin and form of superconductivity, its relation to the correlated insulator, the interplay of topological obstructions and correlations, and the microscopics of their nematic phases. We will work on the impact of spin-orbit coupling and on a theoretical description of twist-angle disorder, viewing inhomogeneities as a blessing in disguise that can also be used to probe and realize interesting physics. Finally, we will develop a theoretical framework for the design of atom arrangements on the surface of complex host materials, in order to create or simulate a quantum many-body system on demand. To this end, we will employ and further extend a variety of analytical and numerical methods of many-body physics and field theory, and combine it, in some projects, with machine-learning techniques, while keeping a close connection to experiment.