Maxwell's equations are a system of vector partial differential equations that model the evolution of electromagnetic waves in a material. The response of the material is encoded in constitutive laws which, in their most general form can be nonlinear, dissipative and dispersive. In this talk, we discuss the construction and analysis of compatible discretizations for the Maxwell system in a variety of linear and nonlinear materials. Such discretizations mimic fundamental properties of the continuous PDEs such as divergence laws and energy relations, providing discrete approximations that are physically meaningful. In this talk, I will discuss past and present work in this area and discuss recent results on energy stable discretizations of Maxwell's equations in nonlinear optical media.