  In the FIRST PART we present simple introductions to gaussian and Bessel
waves, and to the Localized Waves (LW), pulses or beams, showing the important
properties of the latter, and their applications whenever a role is played by a
wave-equation (electromagnetism, optics, acoustics, seismology, geophysics,
gravitation, elementary particle physics,...). The First Part ends with a
historical APPENDIX, recalling how the geometrical methods of Special
Relativity (SR) had predicted the most interesting LWs, i.e., the X-shaped
pulses; and presenting a bird's-eye view of the experiments performed with
evanescent waves (and/or tunnelling photons), and with the "localized
Superluminal solutions". In the SECOND PART, after some more theoretical
introduction, we develop a Generalized "Bidirectional Decomposition", and
obtain several luminal and Superluminal non-diffracting solutions; we get a
space-time focusing of X-Shaped pulses; and deal with chirped optical X-shaped
pulses in material media. Finally, in the THIRD PART we investigate also the
subluminal LWs, which, among the others, allow to emphasize the role of SR, in
its extended, or rather non-restricted, formulation. We study in particular the
topic of zero-speed waves, endowed with a static envelope: Namely, we show how
localized wavefields can be constructed with high transverse localization, and
with a longitudinal intensity pattern that assumes any desired shape within a
chosen interval of the propagation axis. Such "Frozen Waves" promise to have
even more applications. In between, we do not forget to briefly treat the case
of not axially-symmetric solutions, in terms of higher order Bessel beams.
