In this talk, I will first review the history of interior C^2 estimate for fully nonlinear equations. As a matter of fact, only very few equations were shown to have such properties. Even the Monge-Ampere equations for dimension three and higher do not have interior C^2 estimate due to the famous example by Pogorelov.

In the second part, I will discuss my recent work on interior C^2 estimate for Hessian quotient equations. Such equations have deep connections with Monge-Ampere equations, Hessian equations as well as special Lagrangian equations. I will then discuss the main ideas behind the proof. The new method we adopted to prove interior C^2 estimate has independent interest and can be used in other settings.