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Physically accurate and mathematically tractable models are presented to characterize scattering and reflection properties of reconfigurable intelligent surfaces (RISs). We take continuous and discrete strategies to model a single patch and patch array and their interactions with multiple incident electromagnetic (EM) waves. The proposed models consider the effect of the incident and scattered angles, polarization features, and the topology and geometry of RISs. Particularly, a simple system of linear equations can describe the multiple-input multiple-output (MIMO) behaviors of RISs under reasonable assumptions. It can serve as a fundamental model for analyzing and optimizing the performance of RIS-aided systems in the far-field regime. The proposed models are employed to identify the advantages and limitations of three typical configurations. One important finding is that complicated beam reshaping functionality can not be endowed by popular phase compensation configurations. A possible solution is the simultaneous configurations of collecting area and phase shifting. Numerical simulations verify the effectiveness of the proposed configuration schemes.