Hawking and Penrose’s Singularity Theorems are advanced results in General Relativity which describe Big Bang and Black Hole singularities given minimal topological information about the manifold they live in, using the language of Semi-Riemannian Geometry. This talk presents two proofs of the former Theorem: one stronger version which gives a picture of the Big Bang similar to the Robertson-Walker model, and the other which assumes less and proves less. Along the way, we will develop the theory of Focal Points and Index Forms, which play a pivotal role in these proofs, and provide some Physical intuition to justify the Timelike Convergence Condition assumed in Hawking’s Theorem.