This is a book about the biggest subject of all. It’s the ultimate traveller’s guide to everything that could be: the rough and the smooth guide to infinity.Infinity has haunted human minds for thousands of years. It challenges theologians and scientists alike to understand it, cut it down to size, find out if it comes in different shapes and sizes, and decide whether we want to outlaw or welcome it into our human descriptions of the Universe. Is it part of the problem or part of the solution?It is also a live issue. Physicists’ accelerating quest for a Theory of Everything has been primarily guided by an attitude towards infinities. Their appearance can be a warning that you have entered a blind alley on the road to the truth. The enthusiasm with which superstring theories were embraced was a consequence of their ingenious evasion of the problem of infinities that had plagued all their predecessors.These exciting new theories leave us to decide whether we should expect matter to be infinitely divisible. Will we always be able to find ever smaller, more elementary, particles inside any that we have, like a never-ending sequence of Russian dolls? Or is there a limit, a smallest ‘thing’, a smallest size, or a shortest time, where division comes to a full stop? Or perhaps the fundamental entities out of which the world is woven are not really little particles at all?Cosmologists have their own problems with infinities. For decades they have been happy to live with the notion that the Universe of space and time began at a ‘singularity’, where its temperature, its density, and just about everything else, was infinite. But will the marriage of gravity and the quantum really permit actual infinities? Is their appearance a sign of success or failure? Are infinities just a signal that we have not found enough pieces of the puzzle, or are they a vital part of the solution to ultimate problems like the beginning and end of the Universe, the moments of the Big Bang and the…