Abstract: In the introductory first part of the talk, I set up the extension problem for the constraint equations of general relativity. I motivate its study and precise formulation by showing an application to the Cauchy problem in weak regularity. In the second part of the talk, I present my extension procedure which solves the extension problem, and then sketch its proof. In the proof, I use new methods to solve the prescribed divergence equation for a $2$-tensor and the prescribed scalar curvature equation for a metric, both with over-determined boundary conditions.