The TRR 306 QuCoLiMa (Quantum Cooperativity of Light and Matter) is a Collaborative Research Centre Transregio (TRR) of the universities of Erlangen-Nuremberg, Mainz and Saarbrücken. It intends to explore the distinctive traits of quantum cooperativity within a large variety of quantum platforms at the intersection of quantum optics and condensed matter. We aim at understanding what is the interplay of quantum interference and entanglement in the collective response of many-body quantum systems interacting with light. We will explore in particular the role of the quantum properties of radiation in establishing and mediating quantum cooperative phenomena in a variety of complex matter systems, entering the regime of many-body physics of quantum cooperative light-matter.