  We propose a new notion of singularity in General Relativity which
complements the usual notions of geodesic incompleteness and curvature
singularities. Concretely, we say that a spacetime has a volume singularity if
there exist points whose future or past has arbitrarily small spacetime volume:
in particular, smaller than a Planck volume. From a cosmological perspective,
we show that the (geodesic) singularities predicted by Hawking's theorem are
also volume singularities. In the black hole setting, we show that volume
singularities are always hidden by an event horizon, prompting a discussion of
Penrose's cosmic censorship conjecture.
