In this seminar, we discuss the theory of a general material system of N particles coupled to a cavity. We focus on equilibrium. We use bounds for the partition function and a a coherent-state path integral formulation. We obtain the exact (non-local) action wherethe photonic degrees of freedom are replaced by an effective position-dependent interaction between the particles. Besides, in the large-N limit, we show that the theory can be cast into an effective Hamiltonian where the cavity induced interactions are made explicit. We showcase the descriptive power of the formalism with three examples: photon condensation, the 2D free electron gas in a cavity and the modification of magnetic interactions between molecular spins; recovering, condensing and extending some recent results in the literature. We delve into the example of modifying magnetic interactions anticipating the first experimental steps.