Einstein's formulation of gravity in terms of geometry of space and time represented a major paradigm shift from the Newtonian framework, famously predicting, among other things, the final fate of extremely collapsed self-gravitating objects as Black Holes, which, in the eloquent words of S. Chandrasekhar, are “... the most perfect macroscopic objects there are in the universe: the only elements in their construction are our concepts of space and time”. The next key development in General Relativity came through the so called Singularity theorems, first proved by Roger Penrose in 1965 introducing new, powerful tools in Lorentzian geometry, and subsequently generalized by Hawking and Penrose to study spacetime singularities in gravitational collapse and cosmology. In this talk, after giving an elementary introduction to black holes, I will briefly describe the mathematical essence of the singularity theorems, and what they imply for fundamental physics of black holes as well as the origin of our universe. It is a testimony to the profound implications of the deep questions associated with spacetime singularities that the Royal Swedish Academy of Sciences has decided to award the 2020 Nobel prize in Physics jointly to Roger Penrose "for the discovery that black hole formation is a robust prediction of the general theory of relativity", and Reinhard Genzel and Andrea Ghez "for the discovery of a supermassive compact object at the centre of our galaxy"