The performance characterization of communication systems assisted by large reconfigurable intelligent surfaces (RISs) significantly depends on the adopted models for the underlying channels. Under unrealistic channel models, the system performance may be over- or under-estimated which yields inaccurate conclusions for the system design. In this paper, we review five channel models that are chosen to progressively improve the modeling accuracy for large RISs. For each channel model, we highlight the underlying assumptions, its advantages, and its limitations. We compare the system performance under the aforementioned channel models using RIS configuration algorithms from the literature and a new scalable algorithm proposed in this paper specifically for the configuration of extremely large RISs.