In this talk, we investigate mathematical models for electromagnetic wave propagation in dispersive isotropic passive linear media, that is in particular media such as metamterials for which the dielectric permittivity ϵ and magnetic permeability μ depend on the frequency. We emphasize the link between physical requirements and mathematical properties of the models. A particular attention is devoted to the notions of causality and passivity and its connection to Herglotz-Nevanlinna functions. We consider successively the cases of so-called local media and then of general passive media. In particular, we will discuss connections between mathematical properties of models (e.g. stability, “energy conservation”, ...), physical assumptions (e.g. passivity) and the existence of two Herglotz-Nevanlinna functions of the frequency: ωϵ and ωμ that determine the dispersion of the material. We will also present dispersion and spectral analysis of this rather general class of electromagnetic media.