In this talk, we show the regularity, global existence, and convergence of Lagrangian mean curvature flows in the two-convex case . The proof relies on a newly discovered monotone quantity that controls two-convexity of the graphical Lagrangian mean curvature flow. The combination of a blow up argument and a Liouville Theorem for ancient solutions of Lagrangian mean curvature flows is used to prove the convergence of the flow. This is based on a joint work with Chung-Jun Tsai and Mu-Tao Wang.