Penrose’s Weyl Curvature Hypothesis, which dates from the late 70s, is a hypothesis, motivated by observation, about the nature of the Big Bang as a singularity of the space-time manifold. His Conformal Cyclic Cosmology is a remarkable suggestion, made a few years ago and still being explored, about the nature of the universe, in the light of the current consensus among cosmologists (and the Nobel Committee) that there is a positive cosmological constant. I shall review both sets of ideas within the framework of general relativity, emphasise how the second set solves a problem posed by the first, and say something about predictions of CCC.