We prove a number of results on the number of solutions to the asymptotic Plateau problem in 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 Λ in the asymptotic boundary of H^3 , we show that uniqueness of the minimal surfaces with asymptotic boundary Λ is equivalent to uniqueness in the smaller class of stable minimal disks, and, when Λ 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) Λ is the asymptotic boundary of a minimal surface Σ with principal curvatures less than or equal to 1 in absolute value, then uniqueness holds. Contents