**Abstract:** | In this talk, we will present our recent studies about the mean-field interacting particle systems and McKean-Vlasov equation. The contents are fourfold. First we will show that the empirical measure of mean-field model satisfies the large deviation principle with respect to the weak convergence topology or the stronger Wasserstein metric. Second, we will show the gradient estimate of the Poisson equation, the exponential convergence in the Wasserstein metric and uniform in time propagation of chaos for the mean-field particle system related to McKean-Vlasov equation. Third, we will show some explicit and sharp estimates of the spectral gap and the log-Sobolev constant for mean-field particles system, uniform in the number of particles. Last we study the long time behaviour of the kinetic Fokker-Planck equation with mean-field interaction, whose limit is often called Vlasov-Fokker-Planck equation. |