In this talk, we will first recall some sharp geometric inequality and present some new progress on the stability of endpoint case of Sobolev inequality such as Log-Sobolev, Moser-Onofri and Log-HLS . Then we will present our recent progress on the optimal stability for geometric inequality such as Hardy-Littlewood-Sobolev (HLS) inequality, fractional Sobolev inequality, Log-Sobolev inequality in Euclidean space and Heisenberg group. Some interesting open problems will be also discussed.