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.