Engineering design is sometimes inspired by Nature. The natural world is filled with crystals, periodic structures that interact with Schrodinger Waves. Drawing on this analogy, we are designing artificial crystal structures which are intended for Electromagnetic Waves instead. This has now unleashed the collective scientific imagination, engendering a profusion of synthetic electromagnetic crystal structures. In correspondence to semiconductor crystals these usually have an electromagnetic bandgap, a band of frequencies in which electromagnetic waves are forbidden. We will present a pictorial portfolio of various 2 and 3 dimensional crystal structures which have been conceived, and indicate the applications, such as opto-electronic light emitters, radio antennas, and color pigments, for which they are intended. In addition we will emphasize some of the new ideas that will be important for the future, including the further development of radio frequency bandgap structures, and of surface plasmon band structure.