  We prove a number of results on the number of solutions to the asymptotic
Plateau problem in $\mathbb H^3$. In the direction of non-uniqueness, we
construct an example of a quasicircle that is the asymptotic boundary of
uncountably many pairwise distinct stable minimal disks.
  Moreover, we discuss criteria that ensure uniqueness. Given a Jordan curve
$\Lambda$ in the asymptotic boundary of $\mathbb H^3$, we show that uniqueness
of the minimal surfaces with asymptotic boundary $\Lambda$ is equivalent to
uniqueness in the smaller class of stable minimal disks, and, when $\Lambda$ is
invariant by a Kleinian group, to uniqueness in the even smaller class of group
invariant stable minimal disks.
  Finally, we show that if a quasicircle (or more generally, a Jordan curve of
finite width) $\Lambda$ is the asymptotic boundary of a minimal surface
$\Sigma$ with principal curvatures less than or equal to 1 in absolute value,
then uniqueness holds.
