In the last twenty years, cosmology has unexpectedly emerged as one of the most exciting and dynamic fields of modern science. From astoundingly precise measurements of the cosmic microwave background to the ongoing mysteries of dark energy and dark matter, modern cosmology is unquestionably in the midst of its Golden Age. And yet, one of the most eminent mathematical physicists of our age, Roger Penrose is convinced that there is one fundamental problem that is consistently being overlooked: why did our universe begin in such a particular state of extremely low entropy? His Conformal Cyclic Cosmology (CCC) is an attempt to directly address that question. This extensive conversation between Howard Burton and Roger Penrose, co-recipient of the 2020 Nobel Prize in Physics and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute at the University of Oxford, explores his motivation to come up with this theory in the first place and provides detailed insights into his groundbreaking research.This carefully-edited book includes an introduction, A Hidden Audience, and questions for discussion at the end of each chapter:I. Inadvertent Success - An inside look at a mysterious publishing phenomenonII. Considering Entropy - Minimum, maximum and both at the same timeIII. Mysteries and False Explanations - Dark issues and inflationary hyperbole IV. Conformal Geometry - Reinterpreting the end and the beginningV. CCC: The Basic Idea - Matching beginnings and ends, conformally VI. Light Cones - Photons vs. massive particlesVII. Massive Headaches - And how we might make them disappearVIII. Necessary Conditions - Dark energy, dark matter and information loss in black holesIX. Cosmic Imprints? - Searching for evidence of past aeonsX. Battling Onwards - Challenges of an irrepressibly independent thinkerAbout Ideas Roadshow Conversations Series: Presented in an accessible, conversational format, Ideas Roadshow books not only explore frontline academic research featuring world-leading researchers but also reveal the inspirations and personal journeys behind the research.