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 × 10-7 M⊕ (~70 km radius for ρ = 3 g cm-3) could have survived around the transiting planet HD 209458b 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. Earthlike moons of Jovian planets could exist for 5 Gyr in systems where the stellar mass is greater than 0.15 M☉. 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 Qp that a discovery would imply.