The classical singularity theorems of General Relativity show that a Lorentzian manifold with a smooth metric satisfying certain physically reasonable curvature and causality conditions cannot be causal geodesically complete. After a general introduction to the topic we will focus on recent work concerning singularity theorems for metrics that are merely continuously differentiable - a regularity where one still has existence but not uniqueness for solutions of the geodesic equation. I will give an overview of the proof of Hawking's theorem in this regularity and, if time permits, discuss some of the estimates involved in more detail.