this talk is based on a joint work with xiaojun huang entitled “bergman metrics as pull-backs of the fubini-study metric”. we study domains in cn or stein manifolds m such that their bergman metrics have constant holomorphic sectional curvature κ. we prove a uniformization theorem when κ < 0 through the calabi rigidity theorem and holomorphic extension theorems. we also discuss the case when κ ≥ 0. we provide several interesting examples of the existence of such m. under certain conditions on m, we prove that the bergman metric of m can not have non-negative constant holomorphic sectional curvatures.