The classic and pioneering work of Nirenberg, in collaboration with Gidas and Ni, on symmetries of solutions of partial differential equations has generated enormous research activities, and the method of moving planes has become a powerful and user-friendly tool in the study of partial differential equations. In this talk, we will discuss some analytic aspects of second order elliptic or degenerate elliptic conformally invariant equations, in particular on some results where the method of moving planes has played an important role in their proofs. These include some recent joint work with Luc Nguyen on analysis of blow up profiles of solutions of elliptic conformally invariant equations as well as a Bocher type theorem for degenerate elliptic conformally invariant equations, where the method of moving planes has played an important role.