This talk was part of the Workshop on "Mathematical Relativity: Past, Present, Future" held at the ESI December 4 — 7, 2023.

Some theorems in relativity work directly only in low dimensions because of the possibility of singularities in area minimizing hypersurfaces and MOTS. These include both Riemannian and spacetime positive mass theorems as well as the Riemannian Penrose inequality. In some cases the singularities are relatively easy to work around and in others they present significant problems to overcome. This will be a general lecture which will attempt to give perspective and more clarity to this issue.