First I will give a short introduction of the theory of mean field games, which was introduced independently by Lasry and Lions, and Huang, Malhame and Caines to study Nash equilibria for games with a very large number of players. And then I will introduce two long-time behavior results of solutions to evolutionary MFGs system on R^d [P. Cannarsa, W. Cheng, C. Mendico, K. Wang, Dyn. Games Appl., 2020] and with state constraints [P. Cannarsa, W. Cheng, C. Mendico, K. Wang, arXiv: 2004.06505], respectively. At last, I will talk about some new existence results of solutions of discounted stationary MFGs system and of more general (than discounted one) system.