The Einstein equations tie the curvature of a Lorentzian manifold to the matter content of the universe (or vacuum). In this minicourse we first analyze the additional geometric properties needed for a well-posed initial value formulation of the Einstein equations in the standard smooth setting. Real life and the universe, however, need not be everywhere smooth, and so it is important to be able to distinguish features of our theory related to regularity from those related to deeper physical issues, and indeed to understand how they are intertwined. In the case of Lorentzian metrics of low regularity, apart from the obvious fact that curvature is no longer well-defined classically, we will discuss several pathologies and fundamental problems that can arise. We end with a review of recent analytic and geometric approaches that have been developed to effectively handle some of those scenarios.