We discuss possible tests of the constancy of the velocity of light using d
istant astrophysical sources such as gamma-ray bursters (GRBs), active gala
ctic nuclei (AGNs), and pulsars. This speculative quest may be motivated by
some models of quantum fluctuations in the spacetime background, and we di
scuss explicitly how an energy-dependent variation in photon velocity delta
c/c similar to -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 observatio
ns of the GRBs that have recently been identified optically and for which p
recise redshifts are available. We show how a regression analysis can be pe
rformed to look for an energy-dependent effect that should correlate with r
edshift. The present data yield a limit M greater than or similar to 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.