In this expository talk, I will introduce the basic ideas about identifying symplectic quotient and good quotient in the sense of geometric invariant theory. After presenting some finite-dimensional examples, I will discuss the renowned Donaldson-Uhlenbeck-Yau theorem relating slope-stable holomorphic bundle with Hermitian-Yang-Mills connections. If time permits, I will present more applications of the moment map formalism, such as Hitchin's equations and nonabelian Hodge theory.