Abstract: In this talk, I will explain how to combine Fefferman's invariant theory, the Chern-Moser theory, the Cheng-Yau solution of the Fefferman equation, as well as CR extension theory to provide an affirmative solution of a conjecture posed by S. Y. Cheng in 1979. The conjecture stated that the Bergman metric of a bounded strongly pseudoconvex domain is Einstein if and only if the domain is holomorphically equivalent to the ball. This is a joint work with M. Xiao.