An important mathematical subtlety – that is also highly physically relevant – permeating Mathematical General Relativity is the regularity one imposes on – or, from a perhaps more physical point of view, expects of – the metric, the manifold and/or chosen coordinates. In recent years, low regularity analytic methods have become increasingly important in Lorentzian geometry, allowing one to address several longstanding questions. In my talk I will present some of these developments and in particular focus on recent advances regarding the extension of the classical singularity theorems of General Relativity to non-C^2 Lorentzian metrics.