The chapter starts with a study of the Bloch and Fermi varieties associated with a periodic elliptic operator. The Fermi variety (surface) is an analog of the characteristic variety of a constant coefficient operator (i.e., the set of zeros of its symbol). The Bloch variety additionally incorporates the spectral parameter. It thus can be considered as the graph of the multiple-valued function (dispersion relation) from quasi-momenta to the spectrum. The dispersion relation plays a prominent role in the theory of periodic differential equations and all its applications to solid-state physics, photonic crystals, waveguide theory, and other areas. A sometimes-useful incarnation of the Fermi surface, which we call Floquet surface, is also introduced. A study of the geometry of these varieties (including their extensions to the complex domain) is provided, which is critical to resolving many natural questions regarding spectra (as well as other properties) of periodic operators.