We present a generalization of the celebrated results by D. Christodoulou and S. Klainerman for solutions of the Einstein vacuum equations in General Relativity. In 'The global nonlinear stability of the Minkowski space' they showed that every strongly asymptotically flat, maximal, initial data which is globally close to the trivial data gives rise to a solution which is a complete spacetime tending to the Minkowski spacetime at infinity along any geodesic. We consider the Cauchy problem with more general, asymptotically flat initial data. This yields a spacetime curvature which is no longer bounded in $L^{\infty}$. As a major result and as a consequence of our relaxed assumptions, we encounter in our work borderline cases, which we discuss in this talk as well. The main proof is based on a bootstrap argument. To close the argument, we have to show that the spacetime curvature and the corresponding geometrical quantities have the required decay. In order to do so, the Einstein equations are decomposed with respect to specific foliations of the spacetime.