  We investigate the long-term dynamical stability of hypothetical moons
orbiting extrasolar giant planets. Stellar tides brake a planet's rotation and,
together with tidal migration, act to remove satellites; this process limits
the lifetimes of larger moons in extrasolar planetary systems. Because more
massive satellites are removed more quickly than less massive ones, we are able
to derive an upper mass limit for those satellites that might have survived to
the present day. For example, we estimate that no primordial satellites with
masses greater than 7 x 10^{-7} M_Earth$ (~ \70km radius for rho=3 g cm^{-3})
could have survived around the transiting planet HD209458b for the age of the
system. No meaningful mass limits can be placed on moons orbiting Jovian
planets more than ~0.6 AU from their parent stars. Earth-like moons of Jovian
planets could exist for 5 Gyr in systems where the stellar mass is greater than
0.15 M_Sun. Transits show the most promise for the discovery of extrasolar
moons -- we discuss prospects for satellite detection via transits using
space-based photometric surveys and the limits on the planetary tidal
dissipation factor Q that a discovery would imply.
