A brief account of the Cauchy problem in General Relativity

The Cauchy problem in PDE theory has a lot of importance in General Relativity. Some of the prominent aspects are the inclusion of a metric and a given hypersurface to determine the evolution of a given manifold. Some of the aspects we shall discuss in this review are the features of an initial value set in Cosmology, and the significance of some of the results of the Cauchy problem. Our discussion will focus towards stability, singularities, gravitational radiation, and the geometric features of singularities in General Relativity.