In this talk I introduce the Einstein equations, local existence of solutions to them and then motivate their study in low regularity, stating in particular the recent bounded $L^2$-curvature theorem (Klainerman-Rodnianski-Szeftel, 2012). This theorem is proved by Fourier methods and hence assumes initial data to be diffeomorphic to Euclidean $3$-space. I will explain how to localise this result to assuming only data on a compact set with boundary. The proof uses as tools 1.) an extension procedure for the constraint equations, and 2.) the existence of boundary harmonic coordinates on manifolds with boundary.