In the last couple of years much effort has been dedicated to the development and the investigation of hybrid quantum devices, which combine the advantages of very dissimilar quantum systems for the realization of efficient quantum computation and communication technologies. Particularly interesting in this regard are solid-state quantum memories based on ensembles of spins (e.g. nitrogen vacancy defects in diamond or rare-earth doped crystals) coupled to superconducting microwave cavities. Here the spin-ensemble acts as a robust memory where the collective coupling to the cavity mode allows for the coherent transfer of quantum information. The major downside of ensembles inside solid-state systems is their natural tendency to exhibit inhomogeneous broadening of the transition frequencies, which makes experiments as well as their theoretical description very challenging. On the theoretical side it is primarily the computational complexity due to the exponential growth of the associated Hilbert space with the system size that inevitably calls for approximation schemes. To date, the theoretical description of large spin-cavity systems is mainly limited to weak driving fields, where the number of spin-excitations in the ensemble is negligible, corresponding to a Holstein-Primakoff approximation, or to mean-field approaches, where correlations within the system are neglected. In this thesis we aim to develop a model that accurately accounts for the dynamics of large inhomogeneously broadened spin ensembles, coupled to a single cavity mode, even in the case of strong driving fields. In particular we employ the generalized cumulant expansion method to account for correlations within the system, which allows us to go beyond the Holstein-Primakoff and mean-field approximations. As a first step, we demonstrate the applicability of the cumulant expansion technique to the well-studied Jaynes-Cummings model and investigate its main limitations. We then move on to the Tavis-Cummings model with multiple spins inside the cavity and generalize our approach to ensembles containing a very large number of spins (ca. 10^12). Our model aims not only to provide a proper description of ensemble-based quantum memories, but also to serve as a novel tool for the investigation of interesting new physics arising from cooperative effects in inhomogeneous ensembles.