*... to kill that savage monster, the Chimaera, who was not a human being, but a goddess,*

*for she had the head of a lion and the tail of a serpent, while her body was that of a goat, and she breathed forth flames of fire ...*

*(The Iliad, Book VI, translated by Samuel Butler)*

A non-monochromatic superposition of propagation-invariant beams is obviously a propagation-invariant beam itself. However this is no longer the case when the superposition principle ceases to be valid, as in the case of wave propagation in nonlinear media. On the other hand it is well known that the nonlinearity can counteract beam spreading, being at the very origin of those strongly confined propagation-invariant wave-packets that are known as solitons or solitary waves (SW). Unfortunately, SW typically involve a reduced number of dimensions, whereas the observation of a three dimensional (3D) SW has been elusive to date .

Among various reasons behind this, the most relevant are:

(i) the fact that 3D-SW are affected by intrinsic instabilities;

(ii) (more specific to optics) 3D-SW not only require a nonlinear medium (and hence very high laser intensities) but also anomalous material dispersion.

In contrast with SW, localized waves (LW) are 3D propagation-invariant wave-packets that do not rely on any nonlinearity. They have been observed in several contexts and in principle they should not exist in a nonlinear medium since they rely on the superposition principle.

In this respect the experimental observation of optical 3D propagation-invariant pulses, so-called ``light bullets'', in a nonlinear medium with normal dispersion, i.e. in a regime where both LW and SW were not expected to exist, did really appear as an astonishing result.

The spatio-temporal far-field spectrum of the observed light bullets clearly appeared as an X, and an accurate and sophisticated experimental re-construction (``tomography'') of the spatio-temporal profile unveiled the double conical structure of the wave-packet in agreement with theoretical and numerical analysis.

Thus, naturally, the light bullet was dubbed nonlinear X-wave. This result opened many different roads of investigations, mainly aimed at providing for a physical and mathematical background to a special class of nonlinear 3D beams whose features recall both SW (they spontaneously form at high intensity and propagate without sensible distortion) and linear LW (the structure is seemingly that of conical waves), thus constituting a sort of*Chimaera*.

Thereafter the field of nonlinear X-waves (NLX) has rapidly grown, on one hand linking linear LW solutions in dispersive media to the existing literature and the plethora of electromagnetic LW mainly investigated as solutions of Maxwell's equations in vacuum and, on the other hand, attempting to extend the idea of ``non-bell-shaped'' localized solutions to the very active field of nonlinear optics which deals with solitary waves, including periodic media and different kind of nonlinearities. Importantly, numerical simulations, experiments and theory concurred to establish that NLX can be also an effective paradigm for interpreting ultra-fast laser propagation in nonlinear media with an intensity dependent refractive index (i.e., ``Kerr media''). Given the facts that any material displays such a nonlinearity at sufficiently high intensities, and that many experiments have been done in water or air (due to virtual absence of damage threshold), these results have many implications in bio-physical or remote environmental sensing applications.

Moreover, the formal analogy between nonlinear optics in Kerr media and time-evolution of the semi-classical wave-function of ultra-cold bosons has led to the prediction of the existence of ``Matter X-waves'' . Matter waves are a natural manifestation of large scale coherence of an ensemble of atoms populating a fundamental quantum state. The observation of Bose-Einstein condensates (BECs) in dilute ultra-cold alkalis has initiated the exploration of many intriguing properties of matter waves, whose macroscopic behavior can be successfully described via mean-field approach in terms of a single complex wave-function. Large scale coherence effects are usually observed by means of 3D magnetic or optical confining potentials in which BECs are described by their ground-state wave-function. Trapping can also occur in free space (i.e. without a trap) through the mutual compensation of the leading-order (namely, two-body) interaction potential and kinetic energy, leading to matter-SW. This phenomenon, however, has been observed only in 1D. In 2D and 3D, free-space localization cannot occur due to mentioned instability of SW, and even in a trap collapse usually prevents stable formation of BEC, needing stabilizing mechanisms. For these reasons, the use of periodic potentials induced by optical lattices, where the behavior of atoms mimics that of electrons in crystals or photons in periodic media is attracting a great deal of interest: for instance, in 1D (elongated) lattices 1D-SW can form and are referred to as ``gap solitons'' . Notably enough, in the presence of a 1D lattice potential, the 3D dynamics of the fundamental state wave-function formally obeys to the same model of NLX in optics. Specifically, under conditions for which the Bloch state associated with the lattice has a negative effective mass, the natural state of BECs is a localized *matter X wave* characterized by a peculiar bi-conical shape.The atoms are organized in this way in the absence of any trap, solely as the result of the strong anisotropy between the 1D periodic modulation and the free-motion in the 2D transverse plane. In this respect, NLX appear as novel non-trivial localized states of ultra-cold atoms.