I started reading this book when the Proms season began, more than two months ago, and I have to say it proved more challenging than I expected. The book's first sixteen chapters are given over to mathematics; it's not often you find a "popular science" book that discusses hyperbolic geometry in Chapter 2, but defers Schrödinger's Cat until Chapter 29. The earlier mathematical work is straightforward enough, and indeed a refresher for anyone who's done A-level maths, but it rapidly ploughs into new territory. Mostly this is focused on complex numbers and their geometrical properties, constraints and interpretations.
It's quite a relief to move on to the more physically-oriented half of the book. Here Penrose begins by going over standard territory, on relativity and quantum mechanics, though he often chooses a more original - or at any rate, less standard - approach. Penrose is a sceptic of string theory, and prefers his own twistor theory in the elusive search for a "theory of everything". He's careful to warn the reader when expressing a view that is not generally considered part of the current consensus.
My overall impression is that when Penrose discusses topics with which I am broadly familiar, his explanations are lucid and thought-provoking, but I am afraid that when he is on new ground, I find it difficult sometimes to see what he is driving at. He deserves credit for being honest with the reader - by not trying to dumb down by removing all mathematics, but trying to give enough explanation to carry them along. Nevertheless, this is an altogether more difficult book than Hawking's A Brief History of Time, and it's hard to see how it got itself into any bestseller lists. Exercises, relegated to footnotes, seem to imply that this can be used in some contexts as a text book. Though Penrose tries to give the book a conclusion, in many ways it feels as if the book just stops - perhaps this is just as appropriate, as Penrose, though optimistic, is clear that physics is by no means over yet.