On the occasion of Sir Roger Penrose’s 2020 Nobel Prize in Physics, we review
the singularity theorems of General Relativity, as well as their recent
extension to Lorentzian metrics of low regularity. The latter is motivated by
the quest to explore the nature of the singularities predicted by the classical
theorems. Aiming at the more mathematically minded reader, we give a
pedagogical introduction to the classical theorems with an emphasis on the
analytical side of the arguments. We especially concentrate on focusing results
for causal geodesics under appropriate geometric and initial conditions, in the
smooth and in the low regularity case. The latter comprise the main technical
advance that leads to the proofs of $C^1$-singularity theorems via a
regularisation approach that allows to deal with the distributional curvature.
We close with an overview on related lines of research and a future outlook.