  In this chapter, we investigate the mathematical foundation of the modeling
and design of reconfigurable intelligent surfaces (RIS) in both the far- and
near-field regimes. More specifically, we first present RIS-assisted wireless
channel models for the far- and near-field regimes, discussing relevant
phenomena, such as line-of-sight (LOS) and non-LOS links, rich and poor
scattering, channel correlation, and array manifold. Subsequently, we introduce
two general approaches for the RIS reflective beam design, namely
optimization-based and analytical, which offer different degrees of design
flexibility and computational complexity. Furthermore, we provide a
comprehensive set of simulation results for the performance evaluation of the
studied RIS beam designs and the investigation of the impact of the system
parameters.
