Introduction to Quantum-Gravity Phenomenology

After a brief review of the first phase of development of Quantum-Gravity Phenomenology, I argue that this research line is now ready to enter a more advanced phase: while at first it was legitimate to resort to heuristic order-of-magnitude estimates, which were sufficient to establish that sensitivity to Planck-scale effects can be achieved, we should now rely on detailed analyses of some reference test theories. I illustrate this point in the specific example of studies of Planck-scale modifications of the energy/momentum dispersion relation, for which I consider two test theories. Both the photon-stability analyses and the Crab-nebula synchrotron-radiation analyses, which had raised high hopes of beyond-Plankian'' experimental bounds, turn out to be rather ineffective in constraining the two test theories. Examples of analyses which can provide constraints of rather wide applicability are the so-called time-of-flight analyses’', in the context of observations of gamma-ray bursts, and the analyses of the cosmic-ray spectrum near the GZK scale.