  The abstract boundary construction of Scott and Szekeres is a general and
flexible way to define singularities in General Relativity. The abstract
boundary construction also proves of great utility when applied to questions
about more general boundary features of space-time. Within this construction an
essential singularity is a non-regular boundary point which is accessible by a
curve of interest (e.g. a geodesic) within finite (affine) parameter distance
and is not removable. Ashley and Scott proved the first theorem linking
abstract boundary essential singularities with the notion of causal geodesic
incompleteness for strongly causal, maximally extended space-times. The
relationship between this result and the classical singularity theorems of
Penrose and Hawking has enabled us to obtain abstract boundary singularity
theorems. This paper describes essential singularity results for maximally
extended space-times and presents our recent efforts to establish a
relationship between the strong curvature singularities of Tipler and Krolak
and abstract boundary essential singularities.
