The increasing level of experimental control over atomic and optical systems gained in the past years have paved the way for the exploration of new physical regimes in quantum optics and atomic physics, characterised by the appearance of quantum many-body phenomena, originally encountered only in condensed-matter physics, and the possibility of experimentally accessing them in a more controlled manner. In this review article we survey recent theoretical studies concerning the use of cavity quantum electrodynamics to create quantum many-body systems. Based on recent experimental progress in the fabrication of arrays of interacting micro-cavities and on their coupling to atomic-like structures in several different physical architectures, we review proposals on the realisation of paradigmatic many-body models in such systems, such as the Bose-Hubbard and the anisotropic Heisenberg models. Such arrays of coupled cavities offer interesting properties as simulators of quantum many-body physics, including the full addressability of individual sites and the accessibility of inhomogeneous models.
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