Frequency dispersion plays a central role in electromagnetism, and many applications in optics (e.g. filtering devices) are based on it. The basic principles – inertia, causality, passivity – creating and governing frequency dispersion in standard optical media will be presented. Then, the vital role of frequency dispersion in metamaterials will be shown, notably in the cases of the flat lens and invisible systems. Several models of frequency dispersive permittivity will be given, including an extended version of Kramers-Kronig relations and an effective medium description of a multilayered system. In the second part, an augmented formulation of Maxwell equations [A. Tip, Linear absorptive dielectrics, Phys. Rev. A 57 (1998)] will be introduced in order to transform the time-dependent and non-selfadjoint Maxwell operator (with frequency dispersion) into a time-independent and selfadjoint augmented operator. This formulation will be used to show the existence of negative index materials. In addition, a simple frequency linearization procedure, leading to a time-independent but non-selfadjoint operator, will be considered. This linearization has been implemented in the finite element method: complex band structures will be shown for 2D photonic crystals made of Drude metal rods.