The aim of this project is to study non-perturbative realisations of light-matter interaction, achievable in the current experiments in solid-state or circuit QED. The first part of this project is devoted to perform a detailed analysis of the ground and excited states of the theory of dipolar cavity QED in the ultra-strong coupling (USC). Considering advanced numerical techniques, as matrix-product-state based algorithms, I will solve the dynamics of the minimal model representing dipolar cavity QED, given by ^H cQED, for different dipoles configurations. This analysis is crucial in the understanding of how the cavity-induced antiferromagnetism competes with different types of ‘conventional’ spin-ordering, like polarized, Neel-ordered, or frustrated states. The role of frustrated geometries and disorder in the dipolar lattice will be considered in details. The second part of this project consists in develop and analyse a new quantum simulation platform to study USC cavity QED with dipolar systems under fully controlled conditions. To do so I will study systems of polar molecules , which are capacitively coupled to the field of a superconducting LC resonator. The system will completely characterised around the parameters currently achievable in the experiments, with a special attention for the role of dipole-dipole interaction, disorder and driving/dissipation. I will show how one can implement artificially effective USC conditions using Raman schemes, while preserving the natural shape of dipole-dipole interactions in 3D. In this way it will be possible to have a large ensemble of interacting dipoles with a fully controlled coupling to a single cavity mode. Then it will be possible to apply the results acquired from the first part to create a precise map of the quantum simulations of USC experimentally achievable.