  We discuss possible tests of the constancy of the velocity of light using
distant astrophysical sources such as gamma-ray bursters (GRBs), Active
Galactic Nuclei (AGNs) and pulsars. This speculative quest may be motivated by
some models of quantum fluctuations in the space-time background, and we
discuss explicitly how an energy-dependent variation in photon velocity \delta
c/ c \sim - E / M arises in one particular quantum-gravitational model. We then
discuss how data on GRBs may be used to set limits on variations in the
velocity of light, which we illustrate using BATSE and OSSE observations of the
GRBs that have recently been identified optically and for which precise
redshifts are available. We show how a regression analysis can be performed to
look for an energy-dependent effect that should correlate with redshift. The
present data yield a limit M \gsim 10^{15} GeV for the quantum gravity scale.
We discuss the prospects for improving this analysis using future data, and how
one might hope to distinguish any positive signal from astrophysical effects
associated with the sources.
