In complex analysis of one or several variables, one of the motivating questions for research is the classification problem of domains (open, connected sets) up to biholomorphism (bijective, holomorphic maps):  given domains D and D’, when does there exist a bijective holomorphic function from D to D’?  Two tools that are used to study this problem are the Bergman kernel and the Bergman metric.  Metric is in the sense of differential geometry.  In this talk, we will introduce these tools, and in particular, focus on one of the (biholomorphic) invariants associated with the Bergman metric, its holomorphic sectional curvature.   We will give a classification of the bounded domains in C^n with Bergman metrics of constant holomorphic sectional curvature, and we will discuss a generalization of this classification to more general complex manifolds.