The theory of inhomogeneous analytic materials is developed. These are materials where the coefficients entering the equations involve analytic functions. Three types of analytic materials are identified. The first two types involve an integerp. Ifptakes its maximum value, then we have a complete analytic material. Otherwise, it is incomplete analytic material of rankp. For two-dimensional materials, further progress can be made in the identification of analytic materials by using the well-known fact that a 90°rotation applied to a divergence-free field in a simply connected domain yields a curl-free field, and this can then be expressed as the gradient of a potential. Other exact results for the fields in inhomogeneous media are reviewed. Also reviewed is the subject of metamaterials, as these materials provide a way of realizing desirable coefficients in the equations.