Broadly speaking, this research proposal focuses on the equilibrium phases that are or potentially can be realized in strongly interacting electron systems. It can roughly be divided in two main components. The first component is closely related to experiment, and is about a new class of systems called moire superlattices. These systems are van-der-Waals heterostructures of atomically thin two dimensional materials, such as graphene, where the kinetic energy can be quenched in order to obtain interaction-dominated low-energy physics. These moire systems were recently found to host both correlated insulating phases at integer electron fillings, and superconducting phases in between these integer fillings. The goal of the first component is to develop a theoretical understanding of the precise nature of these insulating and superconducting phases, and the mechanisms that give rise to them. The second component of this proposal is not tied to a particular system, but is about developing new simulation methods for general many-electron systems. Concretely, I will combine existing numerical tensor network methods for spin systems with the fermionic tensor network formalism that I developed during my Phd in order to obtain new methods for studying interacting electron systems. In doing so, one of the main guiding principles will be to try to come up with ways to compute experimentally measurable quantities, such as e.g. linear response coefficients.