  An open problem of Manuel Abellanas asks whether every set of disjoint closed
unit disks in the plane can be connected by a conveyor belt, which means a
tight simple closed curve that touches the boundary of each disk, possibly
multiple times. We prove three main results. First, for unit disks whose
centers are both $x$-monotone and $y$-monotone, or whose centers have
$x$-coordinates that differ by at least two units, a conveyor belt always
exists and can be found efficiently. Second, it is NP-complete to determine
whether disks of varying radii have a conveyor belt, and it remains NP-complete
when we constrain the belt to touch disks exactly once. Third, any disjoint set
of $n$ disks of arbitrary radii can be augmented by $O(n)$ "guide" disks so
that the augmented system has a conveyor belt touching each disk exactly once,
answering a conjecture of Demaine, Demaine, and Palop.
