In the first part of this introductory chapter, we present general and formal (simple) introductions to the ordinary Gaussian waves and to the Bessel waves, by explicitly separating the cases of the beams from the cases of the pulses; and, finally, an analogous introduction is presented for the localized waves (LW), pulses or beams. Always we stress the very different characteristics of the Gaussian with respect to the Bessel waves and to the LWs, showing the numerous and important properties of the latter w.r.t. the former ones: Properties that may find application in all fields in which an essential role is played by a wave-equation (like electromagnetism, optics, acoustics, seismology, geophysics, gravitation, elementary particle physics, etc.). In the second part of this chapter (namely, in its Appendix), we recall at first how, in the seventies and eighties, the geometrical methods of special relativity (SR) predicted-in the sense below specified-the existence of the most interesting LWs, i.e., of the X-shaped pulses. At last, in connection with the circumstance that the X-shapedwaves are endowed with superluminal group-velocities (as carefully discussed in the first part of this chapter), we mention briefly the various experimental sectors of physics in which superluminal motions seem to appear: In particular, a bird's-eye view is presented of the experiments till nowperformed with evanescentwaves (and/or tunneling photons), and with the "localized superluminal solutions" to the wave equations. © 2008 John Wiley & Sons, Inc.