General relativity is one of the pillars of modern theoretical physics. Mathematically, it consists in the study of Lorentzian manifolds satisfying the Einstein equations, a system of partial differential equations determining the evolution of the geometry of the manifold. In 1999, Klainerman proposed the bounded $L^2$ curvature conjecture : the initial value problem should be well posed in a of metrics such that the curvature tensor is locally in $L^2$. Following a brief introduction to the study of the Einstein equations, we will present the recent works of Klainerman, Rodnianski and Szeftel establishing the validity of the conjecture. This fundamental result can be seen as the culminant point of a long sequence of works concerning the study of well-posedness for semi-linear and quasilinear wave equations applied to General Relativity and other geometric wave equations.