  Since the early works[1-4] on the so-called nondiffracting waves (called also
Localized Waves), a great deal of results has been published on this important
subject, from both the theoretical and the experimental point of view.
Initially, the theory was developed taking into account only free space;
however, in recent years, it has been extended for more complex media
exhibiting effects such as dispersion[5-7], nonlinearity[8], anisotropy[9] and
losses[10]. Such extensions have been carried out along with the development of
efficient methods for obtaining nondiffracting beams and pulses in the
subluminal, luminal and superluminal regimes[11-18]. This paper (partly a
review) addresses some theoretical methods related to nondiffracting solutions
of the linear wave equation in unbounded homogeneous media, as well as to some
interesting applications of such waves. In section II we analyze the general
structure of the Localized Waves, develop the so called Generalized
Bidirectional Decomposition, and use it to obtain several luminal and
superluminal (especially X-shaped) nondiffracting solutions of the wave
equation. In section III we develop a space-time focusing method by a
continuous superposition of X-Shaped pulses of different velocities. Section IV
addresses the properties of chirped optical X-Shaped pulses propagating in
material media without boundaries. Finally, in Section V, we show how a
suitable superposition of Bessel beams can be used to obtain stationary
localized wave fields, with a static envelope and a high transverse
localization, and whose longitudinal intensity pattern can assume any desired
shape within a chosen interval of the propagation axis.
