The purpose of this research entitled ‘Singularity Theorems in General Relativity’ is to describe elaborately some basic results and applications of global aspects in gravitation and cosmology. It deals with homogeneous, inhomogeneous, isotropic and anisotropic cosmologies and singularities therein. The space-time singularity theorems and the related theoretical advances towards understanding the structure of space-time was first established by Hawking, Penrose and Geroch. The universe is not simply a random collection of irregular distributed matter, but it is a single entity, all parts of which are connected. The existence of space-time singularities indicate the origin of the universe or as the end state of the gravitational collapse of a massive star which has exhausted its nuclear fuel providing the pressure gradient against the inwards pull of gravity. The curvature singularity forming at the end state of collapse, will be completely covered by the event horizon and would be invisible to any external observer.