  We show that the Gromov boundary of the free product of two infinite
hyperbolic groups is uniquely determined up to homeomorphism by the
homeomorphism types of the boundaries of its factors. We generalize this result
to graphs of hyperbolic groups over finite subgroups. Finally, we give a
necessary and sufficient condition for the Gromov boundaries of any two
hyperbolic groups to be homeomorphic (in terms of the topology of the
boundaries of factors in terminal splittings over finite subgroups).
