  These notes summarize a set of lectures on phenomenological quantum gravity
which one of us delivered and the other attended with great diligence. They
cover an assortment of topics on the border between theoretical quantum gravity
and observational anomalies. Specifically, we review non-linear relativity in
its relation to loop quantum gravity and high energy cosmic rays. Although we
follow a pedagogic approach we include an open section on unsolved problems,
presented as exercises for the student. We also review varying constant models:
the Brans-Dicke theory, the Bekenstein varying $\alpha$ model, and several more
radical ideas. We show how they make contact with strange high-redshift data,
and perhaps other cosmological puzzles. We conclude with a few remaining
observational puzzles which have failed to make contact with quantum gravity,
but who knows... We would like to thank Mario Novello for organizing an
excellent school in Mangaratiba, in direct competition with a very fine beach
indeed.
