The solution of electromagnetic field problems requires two basic ingredients: the Maxwell equations and constitutive relations. The purpose of the latter is to provide a description of the electromagnetic properties of the medium or material in which the electromagnetic process (be it propagation, radiation or scattering) takes place. Here the focus is on some of the fundamental aspects of the constitutive characterization of both simple and complex mediums, approached through the dual viewpoints of field analysis in the space-time and the space-frequency domains. Subsequently, the topic is illuminated with a succession of case studies relating to mediums with an ever-increasing structural complexity, ranging from the classical vacuum to bianisotropic and nonlinear mediums. Along the way, important mathematical and physical requirements in the formulation of constitutive relations are encountered. A further focus of this chapter is the emergence of special types of constitutive relations from phenomenological models of materials and via the application of homogenization formalisms.
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