  In this paper we study the geometry and the topology of unbounded domains in
the Hyperbolic Space $\mathbb{H} ^n$ supporting a bounded positive solution to
an overdetermined elliptic problem.
  Under suitable conditions on the elliptic problem and the behaviour of the
bounded solution at infinity, we are able to show that symmetries of the
boundary at infinity imply symmetries on the domain itself.
  In dimension two, we can strengthen our results proving that a connected
domain $\Omega \subset \mathbb{H} ^2$ with $C^2$ boundary whose complement is
connected and supports a bounded positive solution $u$ to an overdetermined
problem, assuming natural conditions on the equation and the behaviour at
infinity of the solution, must be either a geodesic ball or, a horodisk or, a
half-space determined by a complete equidistant curve or, the complement of any
of the above example. Moreover, in each case, the solution $u$ is invariant by
the isometries fixing $\Omega$.
