  Explosive astrophysical transients at cosmological distances can be used to
place precision tests of the basic assumptions of relativity theory, such as
Lorentz invariance, the photon zero-mass hypothesis, and the weak equivalence
principle (WEP). Signatures of Lorentz invariance violations (LIV) include
vacuum dispersion and vacuum birefringence. Sensitive searches for LIV using
astrophysical sources such as gamma-ray bursts, active galactic nuclei, and
pulsars are discussed. The most direct consequence of a nonzero photon rest
mass is a frequency dependence in the velocity of light propagating in vacuum.
A detailed representation of how to obtain a combined severe limit on the
photon mass using fast radio bursts at different redshifts through the
dispersion method is presented. The accuracy of the WEP has been well tested
based on the Shapiro time delay of astrophysical messengers traveling through a
gravitational field. Some caveats of Shapiro delay tests are discussed. In this
article, we review and update the status of astrophysical tests of fundamental
physics.
