  We consider the Klein--Gordon equation associated with the Laplace--Beltrami
operator $\Delta$ on real hyperbolic spaces of dimension $n\!\ge\!2$; as
$\Delta$ has a spectral gap, the wave equation is a particular case of our
study. After a careful kernel analysis, we obtain dispersive and Strichartz
estimates for a large family of admissible couples. As an application, we prove
global well--posedness results for the corresponding semilinear equation with
low regularity data.
