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.