  It is well known that a fundamental theorem of Quantum Field Theory (QFT) set
in at spacetime ensures the CPT invariance of the theory. This symmetry is
strictly connected to the Lorentz covariance, and consequently to the
fundamental structure of spacetime. Therefore it may be interesting to
investigate the possibility of departure from this fundamental symmetry, since
it can furnish a window to observe possible effects of a more fundamental
quantum gravity theory in a "lower energy limit". Moreover, in the past, the
inquiry of symmetry violations provided a starting point for new physics
discoveries. A useful physical framework for this kind of search is provided by
astroparticle physics, thanks to the high energy involved and to the long path
travelled by particles accelerated by an astrophysical object and then revealed
on Earth. Astrophysical messengers are therefore very important probes for
investigating this sector, involving high energy photons, charged particles,
and neutrinos of cosmic origin. In addition, one can also study artificial
neutrino beams, investigated at accelerator experiments. Here we discuss the
state of art for all these topics and some interesting new proposals, both from
a theoretical and phenomenological point of view.
