This chapter deals with a fundamental application of new methods to a geometric quasilinear equation to settle an important conjecture in General Relativity. According to the bounded L² curvature conjecture, the time of existence of a classical solution to the Einstein-vacuum equations depends only on the L²-norm of the curvature and a lower bound on the volume radius of the corresponding initial data set. At a deep level the L² curvature conjecture concerns the relationship between the curvature tensor and the causal geometry of an Einstein vacuum space-time. Thus, though the result is not optimal with respect to the standard scaling of the Einstein equations, it is nevertheless critical with respect to a different scaling tied to its causal properties.