In order to control locally a space-time which satisfies the Einstein equations, what are the minimal assumptions one should make on its curvature tensor? The L2 bounded curvature conjecture roughly asserts that one should only need L2 bound on the curvature tensor on a given space-like hypersuface. This conjecture has its roots in the remarkable developments of the last twenty years centered around the issue of optimal well-posedness for nonlinear wave equations. In this context, a corresponding conjecture for nonlinear wave equations cannot hold, unless the nonlinearity has a very special nonlinear structure. I will report on recent progress towards the proof of the conjecture, which shed light on the specific null structure of the Einstein equations. This is joint work with S. Klainerman and I. Rodnianski.