This talk will be devoted to discuss some recent developments concerning the problem of smoothness of solutions for stationary and parabolic Mean Field Games systems of second order, both in the focusing and defocusing regime. To this aim, we will focus on new regularity aspects in the scale of Lebesgue spaces for viscous Hamilton-Jacobi equations with super-linear gradient growth and unbounded right-hand side, obtained via integral Bernstein methods and duality arguments. We will cover both elliptic and parabolic problems, equipped with various boundary conditions. Finally, we will address the applications of these results to regularity properties of second order Mean Field Games and discuss some open problems.