The analogy between electromagnetic wave propagation in multidimensionally periodic structures and electron wave propagation in real crystals has proven to be a very fruitful one. Initial efforts were motivated by the prospect of a photonic band gap, a frequency band in three-dimensional dielectric structures in which electromagnetic waves are forbidden, irrespective of propagation direction in space. Today many new ideas and applications are being pursued in two and three dimensions, and in metallic, dielectric and acoustic structures, etc. The author reviews the early motivations for this work, which were derived from the need for a photonic band gap in quantum optics. This led to a series of experimental and theoretical searches for the elusive photonic band-gap structures, those three-dimensionally periodic dielectric structures which are to photon waves what semiconductor crystals are to electron waves. Then he describes how the photonic semiconductor can be 'doped', producing tiny electromagnetic cavities. Finally he summarizes some of the anticipated implications of photonic band structure for quantum electronics and the prospects for the creation of photonic crystals in the optical domain.