  In these lectures I review, in as much pedagogical way as possible, various
theoretical ideas and motivation for violation of CPT invariance in some models
of Quantum Gravity, and discuss the relevant phenomenology. Since the subject
is vast, I pay particular emphasis on the CPT Violating decoherence scenario
for quantum gravity, due to space-time foam. In my opinion this seems to be the
most likely scenario to be realised in Nature, should quantum gravity be
responsible for the violation of this symmetry. In this context, I also discuss
how the CPT Violating decoherence scenario can explain experimental
``anomalies'' in neutrino data, such as LSND results, in agreement with the
rest of the presently available data, without enlarging the neutrino sector.
