Let's first dispose, perhaps, of 2 small complaints, which apply to all of John Gribbin's books: John's irrepressible habit to include largely irrelevant biographical data in his texts - as in, in this book, Quote his draft thesis, typed up by his gilrlfriend Nancy Gore, whom he married the following year unquote or "he was born in Washington DC, on 11 November 1930". Frankly - who cares? Another slightly grating habit is the belaboring of extremely elementary points - such as the author's constant reminders of what "10 the power N" means - anyone who would have difficulty grasping this, even if they extraordinarily enough did not know it yet, but nevertheless read popular science books - would surely have got it the first time!
Now for the gist of the book. The book is an overview and analysis of the current state of play in our search for understanding our Universe, either as a unique Universe or as one within a Multiverse of Universes - where our Universe is one of many (a more technical, and in some ways narrower, overview of learned opinions on the subject ranging from strong acceptance to strong rejection of the concept(s) of the Multiverse is to be found in the book 'Universe or Multiverse, edited by Bernard Carr)
John Gribbin's book shines in many ways, but leaves some questions hanging and IMHO does not go far enough in certain areas. Commendably, he cites Edward Tryon's work - a work that had been rejected out of hand by many eminent Physicists, because Tryon was way ahead of his time when he first described in the late sixties our Universe as the possible result of a rogue quantum fluctuation in a pre-existing environment. The reason for the rejection was that the inflationary scenario (as put forward by Alan Guth) was not yet understood - yet, when I discussed Tryon's model with a couple of world-renowned Physicists as recently as 2005, several years after Alan Guth became famous, they still rejected Tryon's ideas out of hand.
A couple of points that are mentioned almost in passing by John Gribbin would require book-length treatment, and some meta-results seem assumed rather than proven. For instance, he commendably indicates, almost in passing, that time is quantized (an idea astonishingly still controversial in some quarters) and without further ado sets the value of the time quantum at the Planck value. There is absolutely no evidence that the time quantum indeed has that value - the Planck time solely sets an upper boundary to a range of possible time quantum values - there is most likely one time quantum value per Universe within the Metaverse.
Finally- Max Tegmark is a well-known proponent of mathematics as being the ultimate reality - and although John Gribbing cites Max Tegmark's work several times, and in addition rightly says in the course of the text that 'the truth lies in the equations', he does not explore enough the explanatory and predictive power that pure mathematics lends our attempts to explain the Universe.
As for the conclusion - no spoiler here - I am a whit worried that the conclusion does not address properly an issue it raises, that of backwards recurrence. Overall, a five-star effort, possibly better read in conjunction with Bernard Carr's compilatory volume, but an excellent book in its own right.