  Good clocks are of importance both to fundamental physics and for
applications in astronomy, metrology and global positioning systems. In a
recent technological breakthrough, researchers at NIST have been able to
achieve a stability of 1 part in $10^{18}$ using an Ytterbium clock. This
naturally raises the question of whether there are fundamental limits to the
stability of clocks. In this paper we point out that gravity and quantum
mechanics set a fundamental limit on the stability of clocks. This limit comes
from a combination of the uncertainty relation, the gravitational redshift and
the relativistic time dilation effect. For example, a single ion hydrogen maser
clock in a terrestrial gravitational field cannot achieve a stability better
than one part in $10^{22}$. This observation has implications for laboratory
experiments involving both gravity and quantum theory.
