The fundamental laws of physics, as we presently understand them, depend
on about 25 parameters, such as Planck's constant h, the gravitational
constant G, and the mass and charge of the electron.  It is natural to
ask whether these parameters are really constants, or whether they vary
in space or time.

Interest in this question was spurred by Dirac's large number
hypothesis.  The "large number" in question is the ratio of the
electric and the gravitational force between two electrons, which is
about 10^40; there is no obvious explanation of why such a huge number
should appear in physics.  Dirac pointed out that this number is
nearly the same as the age of the Universe in atomic units, and
suggested in 1937 that this coincidence could be understood if
fundamental constants---in particular, G---varied as the Universe
aged.  The ratio of electromagnetic and gravitational interactions
would then be large simply because the Universe is old.  Such a
variation lies outside ordinary general relativity, but can be
incorporated by a fairly simple modification of the theory.  Other
models, including the Brans-Dicke theory of gravity and some versions
of superstring theory, also predict physical "constants" that vary.

Over the past few decades, there have been extensive searches for
evidence of variation of fundamental "constants."  Among the methods
used have been astrophysical observations of the spectra of distant
stars, searches for variations of planetary radii and moments of
inertia, investigations of orbital evolution, searches for anomalous
luminosities of faint stars, studies of abundance ratios of radioactive
nuclides, and (for current variations) direct laboratory measurements.

One powerful approach has been to study the "Oklo Phenomenon," a uranium
deposit in Gabon that became a natural nuclear reactor about 1.8 billion
years ago; the isotopic composition of fission products has permitted a
detailed investigation of possible changes in nuclear interactions.
Another has been to examine ratios of spectral lines of distant quasars
coming from different types of atomic transitions (resonant, fine
structure, and hyperfine).  The resulting frequencies have different
dependences on the electron charge and mass, the speed of light, and
Planck's constant, and can be used to compare these parameters to their
present values on Earth.  Solar eclipses provide another sensitive test
of variations of the gravitational constant.  If G had varied, the
eclipse track would have been different from the one we calculate today,
so the mere fact that a total eclipse occurred at a particular location
provides a powerful constraint, even if the date is poorly known.

So far, these investigations have found no evidence of variation of
fundamental "constants."  The current observational limits for most
constants are on the order of one part in 10^10 to one part in 10^11 per
year.  So to the best of our current ability to observe, the
fundamental constants really are constant.

References: 

For a good short introduction to the large number hypothesis and the
constancy of G, see:

  C.M. Will, _Was Einstein Right?_ (Basic Books, 1986)

For more technical analyses of a variety of measurements, see:

  L. L. Cowie & A. Songaila, Astrophysical Journal (1995) v. 453,
       p. 596 also available online at
       <URL:
       http://adsabs.harvard.edu/cgi-bin/nph-article_query?1995ApJ...453..596C>

  P. Sisterna & H. Vucetich, Physical Review D41 (1990) 1034 and
     Physical Review D44 (1991) 3096

  E.R. Cohen, in _Gravitational Measurements, Fundamental Metrology and
     Constants_, V. De Sabbata & V.N. Melnikov, editors (Kluwer
     Academic Publishers, 1988)

  "The Constants of Physics," Philosophical Transactions of the Royal
     Society of London A310 (1983) 209--363