Abstract
This chapter discusses the experimental and theoretical progress in understanding the properties of photonic band-gap (PBG) materials. Because PBG materials are generally artificially made structures, there are an infinite number of possible PBG materials depending on the crystal structure, the dimensionality, the lattice parameter, the filling fraction, and the dielectric constants of the constituent materials. The complication for PBG crystals compared to electronic crystals is that Maxwell's equations describing the propagation of electromagnetic waves are vectorial in nature, while the Schroedinger equation describing electronic waves is scalar. It is this feature that can give rise to nontrivial generalizations of the well-known properties for electrons in solids. The chapter also presents a vector formalism for photonic properties in PBG materials that parallels the well-known scalar formalism for electronic properties in solids.