  We consider non-scattering energies and transmission eigenvalues of compactly
supported potentials in the hyperbolic spaces $\mathbb H^n$. We prove that in
$\mathbb H^2$ a corner bounded by two hyperbolic lines intersecting at an angle
smaller than $180^\circ$ always scatters, and that one of the lines may be
replaced by a horocycle. In higher dimensions, we obtain similar results for
corners bounded by hyperbolic hyperplanes intersecting each other pairwise
orthogonally, and that one of the hyperplanes may be replaced by a horosphere.
The corner scattering results are contrasted by proving discreteness and
existence results for the related transmission eigenvalue problems.
