  We present several local and global results on isometric immersions of
Kaehler manifolds $M^{2n}$ into hyperbolic space $\Hy^{2n+p}$. For instance, a
classification is given in the case of dimension $n\geq 4$ and codimension
$p=2$. Moreover, as corollaries of general results, we conclude that there are
no isometric immersion in codimension $p\leq n-2$ if the Kaehler manifold is of
dimension $n\geq 4$ and either has a point of positive holomorphic sectional
curvature or is compact.
