The original singularity theorems of Penrose and Hawking have, in their hypotheses, pointwise energy conditions violated by some classical and all quantum fields. If we want to extend their validity to semiclassical gravity, these conditions have to be replaced by weaker ones. In this talk I will first discuss recent results for singularity theorems with weakened energy conditions, some of which are obeyed by quantum fields. Then I will argue for the need of singularity theorems with worldvolume averaged energy conditions both in the timelike and the null case. For each case I will present progress and open questions.

The classical singularity theorems predict the existence of singularities, defined using incomplete geodesics, under a set of general assumptions. One of those assumptions, namely the energy condition, is always violated by quantum fields and thus the realm of semiclassical gravity is outside the scope of these theorems. However, quantum fields do obey weaker conditions which can also be used to predict singularities. In this talk, I will present derivations of such semiclassical singularity theorems both in the timelike and the null case and discuss the challenges and open questions for each case.