This book introduces Conformal Cyclic Cosmology: an amazingly beautiful idea, which I would love to be true. Unfortunately, the evidence to date is far from compelling. But, even if it isn't correct, Penrose is asking such interesting questions that the book is absolutely worth reading.
So here's my understanding of what it's about. Penrose starts by explaining the basic puzzle, which was a key theme in The Road to Reality
and has been tantalizing cosmologists in general for a good while. As everyone knows who was paying attention during high school physics, the Second Law of Thermodynamics says that entropy increases with time; the universe gets progressively more and more disordered. If you drop an egg off a table, it breaks. But the time-reverse of this process is so vanishingly improbable that you don't ever expect to see it happen. A mess of egg-white, yolk and shell will not magically reassemble itself into an intact egg and then bound off the floor onto a table.The rest of this review is in my book If Research Were Romance and Other Implausible Conjectures