Abstract: We study the fully nonlinear elliptic equations in several complex variables and plurisubharmonic solutions to them. We first present a constant rank theorem for a Hessian matrix of plurisubharmonic solution to some differential equation, which has important applications in complex analysis and complex geometry. We will also present a joint work with P. Guan and X. Zhang on the uniqueness of some extremal metrics on K\"ahler manifolds. Finally we will discuss the on-going work on complex homogeneous Monge-Amp\`ere equations if time permits.