In this talk I will present recent work on the rigorous justification of the geometric optics approximation for the Einstein vacuum equations, and its link with the Burnett conjecture in general relativity. I will start by presenting the initial value problem for the Einstein vacuum equations formulated in wave coordinates. Then I will give the state of the art on the Burnett conjecture, focusing on the approaches in U(1) symmetry and double null gauge. I will then present my main result and sketch its proof, highlighting the quasi- and semi-linear challenges. I will conclude my talk by discussing other quadratic wave equations.