This proposal shows a new path to explore frontiers in quantum many-body physics using degenerate atomic gases. We will address fundamental open questions, create novel quantum-many body systems and seek applications beyond the realm of quantum gases. A two-component Fermi gas in an optical lattice is a unique realisation of the Fermi-Hubbard model and it is intimately linked to elementary concepts and open questions in many-body physics. We will develop novel tools for continuous cooling and detection of fermionic atoms in optical lattices. This will enable us to enter the anti-ferromagnetic phase and to study fundamental questions concerning the interplay between localization, coherence and spin-ordering in quantum many-body systems. An intriguing direction towards the creation of novel quantum many-body systems is the coupling of a strongly correlated quantum gas to an optical cavity. Here the cavity creates an effective long-range interaction with global character. This will bring together the physics of strongly-correlated systems and non-linear phenomena using a microscopically accessible system. In this highly explorative field we envisage, as a first experiment, a study of cavity-driven self-organization which may allow us to identify a novel form of a supersolid phase. Rather than investigating or manipulating the quantum gas using light we will also invert this approach and study the light after the interaction with a quantum gas inside a cavity. Using cavity opto-mechanical effects and a van der-Waals blockade by Rydberg atoms excited inside the cavity we will explore squeezing of the light and a novel photon blockade.