  We construct the fundamental solution of the Porous Medium Equation posed in
the hyperbolic space $H^n$ and describe its asymptotic behaviour as
$t\to\infty$. We also show that it describes the long time behaviour of
integrable nonnegative solutions, and very accurately if the solutions are also
radial and compactly supported. By radial we mean functions depending on the
space variable only through the geodesic distance $r$ from a given point $O\in
H^n$. We also construct an exact generalized traveling wave solution. We show
that the location of the free boundary of compactly supported solutions grows
logarithmically for large times, in contrast with the well-known power-like
growth of the PME in the Euclidean space. Very slow propagation at long
distances is a feature of porous medium flow in hyperbolic space.
