Three-dimensionally periodic dielectric structures, photonic crystals, possessing a forbidden gap for electromagnetic wave propagation, a photonic band gap, are now known. If the perfect 3D periodicity is broken by a local defect, local electromagnetic modes can occur within the forbidden band gap. Addition of extra dielectric material locally, inside the photonic crystal, produces ``donor'' modes. Conversely, local removal of dielectric material from the crystal produces ``acceptor'' modes. It is now possible to make high-Q electromagnetic cavities of \ensuremath{\sim}1 cubic wavelength, for short wavelengths at which metallic cavities are useless. These new dielectric cavities can cover the range from mm waves to UV wavelengths.