This thesis reports on the realization of the first experiments conducted with superfluid, strongly interacting Fermi gases of 6Li coupled to the light field of an optical cavity. In the scope of existing ultracold atomic platforms, this is the first time that a system with strong ground state fermionic correlations is operated in the framework of cavity quantum electrodynamics (cQED). From a condensed matter perspective, the system features a fully controllable microscopic Hamiltonian with control over both the strength of the ground state and light-matter interactions and the geometry of the latter. This contrasts with usual solid-state systems, in which the properties of the ground states are hardly tunable. As such our experiment is the perfect platform to simulate the physics of strongly correlated matter coupled to light fields. The manuscript is divided in three parts. The first part is dedicated to the presentation of technical details of the experiment and of measurement techniques we routinely employ to create and probe our strongly interacting gases coupled to light. We introduce the use of the cavity as a probing tool by presenting the achievement of strong light-matter coupling between the atomic ensemble and the cavity field. Similarly, we present a robust thermometry technique for the unitary Fermi gas, with which we measure temperature of the gases deep in the superfluid, quantum degenerate regime. In a second part we focus on the measurements of the strong atom-atom correlations which emerge from the energy spectrum of the atoms-cavity system. We start by laying down the theoretical basis needed for the understanding of the origin of the strong atom-atom interaction, and present its consequence on the many-body wavefunction of the gas by introducing the two-body contact as a universal thermodynamic quantity. We then report on the observation of strong light-matter coupling between pairs of atoms and the cavity field via photoassociation transitions. We describe the resulting light-matter coupling strength in terms of the two-body contact, imprinting many-body correlations onto cavity spectra for the first time. In the following experiment we study the optomechanical response of the gas in the dispersive regime. We observe distorted cavity transmission profiles, signatures of the nonlinear Kerr effect. The strength of the nonlinearity is governed by the density response of the gas, which we express via an operator product expansion in terms of the contact. In the last part, we investigate the effects of engineering long-range, photon-mediated interactions in the gas. We formally show how the system is expected, above a critical value for the strength of the long-range interaction, to undergo a phase transition to a density ordered state. The onset of density-ordering is observed by the superradiant properties of the ordered phase, and we show that it is also controlled by the density response of the unperturbed gas. In addition, we measure the divergence of the density wave susceptibility as the strength of the long-range interactions approaches the critical points: a striking feature of phase transitions. By measuring its temperature after the experiment, we prove that the gas remains superfluid.