In this talk I first set up the Cauchy problem of general relativity and introduce the extension problem for initial data, motivated by applications to the Cauchy problem. Then I proceed to solve the extension problem, that is, I show that small initial data on the unit ball can be extended to asymptotically flat initial data on the full Euclidean space. For the proof, I use new methods to solve the prescribed divergence equation for the second fundamental form and the prescribed scalar curvature equation for the metric. I use the under-determinedness of the constraint equations to preserve regularity.