  We review recent work on the existence and nature of cosmological
singularities that can be formed during the evolution of generic as well as
specific cosmological spacetimes in general relativity. We first discuss
necessary and sufficient conditions for the existence of geodesically
incomplete spacetimes based on a tensorial analysis of the geodesic equations.
We then classify the possible singularities of isotropic globally hyperbolic
universes using the Bel-Robinson slice energy that closely monitors the
asymptotic properties of fields near the singularity. This classification
includes all known forms of spacetime singularities in isotropic universes and
also predicts new types.
