In this talk we present a structural condition on non-separable Hamiltonians, which we termed displacement monotonicity condition, to study second order mean field games master equations. A rate of dissipation of a bilinear form is brought to bear a global (in time) well-posedness theory, based on a-priori uniform Lipschitz estimates on the solution in the measure variable. Displacement monotonicity being sometimes in dichotomy with the widely used Lasry-Lions monotonicity condition, the novelties of this work persist even when restricted to separable Hamiltonians. Our results have been obtained in collaboration with W. Gangbo, C. Mou and J. Zhang.