  Each of the giant planets within the Solar System has large moons but none of
these moons have their own moons (which we call ${\it submoons}$). By analogy
with studies of moons around short-period exoplanets, we investigate the
tidal-dynamical stability of submoons. We find that 10 km-scale submoons can
only survive around large (1000 km-scale) moons on wide-separation orbits.
Tidal dissipation destabilizes the orbits of submoons around moons that are
small or too close to their host planet; this is the case for most of the Solar
System's moons. A handful of known moons are, however, capable of hosting
long-lived submoons: Saturn's moons Titan and Iapetus, Jupiter's moon Callisto,
and Earth's Moon. Based on its inferred mass and orbital separation, the
newly-discovered exomoon candidate Kepler-1625b-I can in principle host a large
submoon, although its stability depends on a number of unknown parameters. We
discuss the possible habitability of submoons and the potential for
subsubmoons. The existence, or lack thereof, of submoons, may yield important
constraints on satellite formation and evolution in planetary systems.
