One of the grand promises of van der Waals heterostructures, especially of the latest wave of twisted graphene systems, are novel phases of matter driven by electronic interactions. Correspondingly, the goal of this research project is to enhance our knowledge about interaction effects in these systems: (i) we want to explore the rich zoo of twisted van der Waals hetero-structures beyond the bi- and multi-layer graphene paradigm on the level of effective low-energy or few-band models obtained from density functional theory characterizations as input to unbiased methods for the study of correlations. (ii) We extend our study to incorporate the entirety of the large moiré Brillouin zone, explicitly keeping a larger span of high-energy degrees of freedom in the description. This allows us to make predictions that cannot be found considering the bands near the Fermi level alone. This second approach, again fed with ab-initio input obtained by collaborating theory groups in this priority program, gives us a chance to develop quantitatively controlled results that are not potentially hampered by complicated strong-interaction effects that occur easily in studies restricted to the few lowest bands.