  The Poincar\'e relativity principle has been tested at low energy with great
accuracy, but its extrapolation to very high-energy phenomena is much less well
established. Lorentz symmetry can be broken at Planck scale due to the
renormalization of gravity or to some deeper structure of matter: we expect
such a breaking to be a very high energy and very short distance phenomenon. If
textbook special relativity is only an approximate property of the equations
describing a sector of matter above some critical distance scale, an absolute
local frame (the "vacuum rest frame", VRF) can possibly be found and
superluminal sectors of matter may exist related to new degrees of freedom not
yet discovered experimentally. The new superluminal particles ("superbradyons",
i.e. bradyons with superluminal critical speed) 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 (c_i >> c where c_i is the critical speed of a superluminal
sector). They may be the ultimate building blocks of matter At speed v > c,
they are expected to release "Cherenkov" radiation ("ordinary" particles) in
vacuum. Superluminal particles could provide most of the cosmic (dark) matter
and produce very high-energy cosmic rays. 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. Very large volume and unprecedented background rejection ability are
crucial requirements for any detector devoted to the search for cosmic
superbradyons. Future cosmic-ray experiments using air-shower detectors
(especially from space) naturally fulfil both requirements.
