Spacetime Singularities

We present here an overview of our basic understanding and recent developments on spacetime singularities in the Einstein theory of gravity. Several issues related to physical significance and implications of singularities are discussed. The nature and existence of singularities are considered which indicate the formation of super ultra-dense regions in the universe as predicted by the general theory of relativity. Such singularities develop during the gravitational collapse of massive stars and in cosmology at the origin of the universe. Possible astrophysical implications of the occurrence of singularities in the spacetime universe are indicated. We discuss in some detail the profound and key fundamental issues that the singularities give rise to, such as the cosmic censorship and predictability in the universe, naked singularities in gravitational collapse and their relevance in black hole physics today, and their astrophysical implications in modern relativistic astrophysics and cosmology.Comment: 45 pages, LaTex; Invited Review article for the `Springer Handbook of Spacetime' (eds A. Ashtekar and V. Petkov