The theory of inhomogeneous analytic and polynomial materials is developed. These are media where the coefficients entering the equations involve analytic functions or polynomials. Three types of analytic or polynomial materials are identified. The first two types involve an integer p. If p takes its maximum value then we have a complete analytic or polynomial material. Otherwise it is incomplete analytic or polynomial material of rank p. For two-dimensional materials further progress can be made in the identification of analytic and polynomial materials by using the well-known fact that a 90 degrees 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.