With Bun Wong at UC Riverside, we study bounded domains in C^n with the Bergman metric of constant holomorphic sectional curvature. We give equivalent conditions for the domains being biholomorphic to a ball in terms of the exhaustiveness of the Bergman-Calabi diastasis. In particular, we prove that such domains are Lu Qi-Keng. We also extend a theorem of Lu towards the incomplete situation and characterize pseudoconvex domains that are biholomorphic to a ball possibly less a relatively closed pluripolar set. Time permitting, I will mention some results on Riemann surfaces.