Nash equilibria for N-player stochastic differential games in closed-loop strategies are described by strongly coupled systems of N Hamilton–Jacobi–Bellman–type equations, known as Nash systems. Obtaining regularity estimates for such systems that behave well as N grows is a crucial step in the study of large-population games, in order to justify limit models as N goes to infinity. I will present some recent results in this direction and discuss their implications for mean field game theory and related, more general frameworks. The talk is based on joint works with Marco Cirant (Padova) and Joe Jackson (Chicago).