  Evidence on the violation of Lorentz symmetry arises from the observation of
cosmic rays with energies beyond the GZK cutoff, $E_{GZK} \simeq 4 \times
10^{19} eV$, from the apparent transparency of the Universe to the propagation
of high energy gamma radiation and from the stability of pions in air showers.
These three paradoxes can be explained through deformations of the relativistic
dispersion relation. Theoretical ideas aimed to understand how Lorentz symmetry
may be broken and phenomenologically interesting deformations of the
relativistic dispersion relation may arise are briefly discussed.
