Abstract: Efforts to place limits on deviations from canonical
formulations of electromagnetism and gravity have probed length scales increasing
dramatically over time.Historically, these studies have passed through three
stages: (1) Testing the power in the inverse-square laws of Newton and
Coulomb, (2) Seeking a nonzero value for the rest mass of photon or graviton, (3)
Considering more degrees of freedom, allowing mass while preserving
explicit gauge or general-coordinate invariance. Since our previous review the
lower limit on the photon Compton wavelength has improved by four orders of
magnitude, to about one astronomical unit, and rapid current progress in
astronomy makes further advance likely. For gravity there have been vigorous
debates about even the concept of graviton rest mass. Meanwhile there are
striking observations of astronomical motions that do not fit Einstein
gravity with visible sources. "Cold dark matter" (slow, invisible classical
particles) fits well at large scales. "Modified Newtonian dynamics" provides the
best phenomenology at galactic scales. Satisfying this phenomenology is a
requirement if dark matter, perhaps as invisible classical fields, could be
correct here too. "Dark energy" {\it might} be explained by a
graviton-mass-like effect, with associated Compton wavelength comparable to the radius
of the visible universe. We summarize significant mass limits in a table.