On the localized superluminal solutions to the Maxwell equations

The various experimental sectors of physics in which superluminal motions seem to appear are briefly mentioned. In particular, a birdʹs-eye view is presented of the experiments with evanescent waves (and/or tunneling photons) and with the "localized superluminal solutions" (SLS) to the wave equation, like the so-called X-shaped beams. The authors also present a series of new SLSs to the Maxwell equations, suitable for arbitrary frequencies and arbitrary bandwidths, some of them endowed with finite total energy. Among the others, the authors set forth an infinite family of generalizations of the classical X-shaped wave and show how to deal with the case of a dispersive medium. Results of this kind may find application in other fields in which an essential role is played by a wave equation (like acoustics, seismology, geophysics, gravitation, elementary particle physics, etc.).