  Lorentz symmetry has been tested at low energy with great accuracy, but its
extrapolation to very high-energy phenomena is much less well established. We
expect a possible breaking of Lorentz symmetry to be a very high energy and
very short distance phenomenon, compatible with existing data. If textbook
special relativity is only an approximate property of the equations describing
a sector of matter above some critical distance scale, superluminal sectors of
matter may exist related to new degrees of freedom not yet discovered
experimentally. The new superluminal particles ("superbradyons") would have
positive mass and energy, and behave kinematically like "ordinary" particles
(those with critical speed in vacuum equal to c , the speed of light) apart
from the difference in critical speed (we expect c_i >> c , where c_i is the
critical speed of a superluminal sector of matter). At speed v > c , they are
expected to release "Cherenkov" radiation ("ordinary" particles) in vacuum. If
superluminal particles exist, they could provide most of the cosmic (dark)
matter and produce very high-energy cosmic rays compatible with unexplained
discoveries reported in the literature. We discuss: a) the possible relevance
of superluminal matter to the composition, sources and spectra of high-energy
cosmic rays; b) signatures and experiments allowing to possibly explore such
effects.
