  In this paper we study supercritical super-OU processes with general
branching mechanisms satisfying a second moment condition. We establish central
limit theorems for the super-OU processes. In the small and crtical branching
rate cases, our central limit theorems sharpen the corresponding results in the
recent preprint of Milos in that the limit normal random variables in our
central limit theorems are non-degenerate. Our central limit theorems in the
large branching rate case are completely new. The main tool of the paper is the
so called "backbone decomposition" of superprocesses.
