Einstein's general theory of relativity provides a geometrical description of gravity in terms of space-time curvature. The Einstein field equations pose fascinating challenges that have stimulated a great deal of research in geometry and partial differential equations. Important questions include the well-posedness of the initial value problem, the linear and non-linear stability of space-times, the formation of black holes, and the boundary value problems arising from the classical aspects of the AdS/CFT correspondence. I will give a survey of some significant advances and open problems pertaining to these questions. No background knowledge of General Relativity will be assumed.