  We study the theory of convergence for CAT$(0)$-lattices (that is groups
$\Gamma$ acting geometrically on proper, geodesically complete CAT$(0)$-spaces)
and their quotients (CAT$(0)$-orbispaces). We describe some splitting and
collapsing phenomena, explaining precisely how these action can degenerate to a
possibly non-discrete limit action. Finally, we prove a compactness theorem for
the class of compact CAT$(0)$-homology orbifolds, and some applications: an
isolation result for flat orbispaces and an entropy-pinching theorem.
