Ultracold atoms in an optical lattice constitute a clean, controlled and tunable implementation of many intractable models in condensed matter physics. In the long run, cold gas experiments may act as quantum simulators and shed new light on those models, and tell us to what extent the real material is described by the model. The prerequisite is, however, that those experiments are benchmarked and validated against models with known results. A lot of recent attention was therefore devoted to the quantitative analysis of these experiments: we have shown for instance that interference patterns for bosonic gases in an optical lattice can be in one-to-one agreement with quantum Monte Carlo simulations. In recent years, as the many-body physics has become more apparent, the numerical component in explaining cold gas experiments has grown stronger. Unfortunately, no numerical methods exist today that can control the answer in the new parameter regimes that gradually become accessible in cold gas experiments.In this proposal we want to develop novel diagrammatic quantum Monte Carlo methods aiming at controlled answers for intractable models in parameter regimes relevant for cold atoms. By a combination of cluster dynamical mean-field theory and diagrammatic Monte Carlo we will investigate the Hubbard model in the pseudo-gap regime. For fermions with long-range interactions such as dipolar and Coulombic systems, diagrammatic Monte Carlo will also be developed. This can have far reaching consequences for high Tc superconductors, exchange functionals and material science. Sideprojects in this proposal include the study of spinor bosonic systems in higher dimensions, bosonic polar molecules with anisotropic interactions in different geometries, polaronic effects in cold gases, and circuit quantum electrodynamics, where the strong light-matter coupling opens a different avenue for quantum simulation, but with its own challenges.