The book is an exploration of the development of Tegmark's leading edge (or wacky, depending on your point of view) ideas - I should stress, though, whether or not he's right, Tegmark is a respected physicist, not a random person with no knowledge to back up his ideas. The book includes an excellent pass through the development of the current hot big bang with inflation theory that it would be worth buying for without the rest. In his introduction, Tegmark says that regular popular science readers might want to skip these first few chapters, but I really recommend that you don't - for instance, he gives the best explanation and exploration of the concept of inflation I've ever seen in a popular science book. It's superb.
From then on, though I don't necessarily accept what Tegmark has to say, he gives a very engaging picture of the way that the concept of eternal inflation could produce a multiverse with a infinite collection of big bangs, each producing their own universe, an impassioned plea for the many worlds interpretation of quantum theory and a really impressive attempt to persuade us that the universe isn't just described by mathematics, but is fundamentally mathematical at its heart.
To be honest, you can stop there and go and buy it if you like. But I do have to say why, personally, I'm not very convinced by anything Tegmark says once he leaves the mainstream. I also have a couple of niggles about the book, which I'll get out of the way first. I found the bits about his personal life more distracting than helpful (though I know publishers love this kind of thing). He several times refers to the detailed colour picture of the Cosmic Microwave Background radiation that is shown on the cover. As you can see from the image above, if this is correct, then the universe is a whole lot more interesting that I thought it was. It's a shame the text wasn't updated to reflect the new edition. Also, the BICEP2 results form quite a big piece of evidence in favour of his view of inflation - unfortunately the book seems to have been published just before these were effectively dismissed, which would put Tegmark's reflections on BICEP in a very different light.
I won't spend too much on what I wasn't convinced by in the content, but a few key points are that he makes several deductions from infinity which I don't think can be justified (you have to be very careful, deducing things from infinity), for all his enthusiasm I wasn't sold on the many worlds interpretation of quantum mechanics, and I think in the final part of the book he makes the common error of conflating models and reality.
However, as I mentioned up front, I didn't care - because even when I didn't agree with him, I found the book really made me think. Which surely is a mark of class.
Review by Brian Clegg