Who has not wondered if there is something beyond the universe? Or if there is a biggest number? Or if there is something smaller than the smallest thing? These are questions about different infinities. Exploring the paradoxes and riddles of what infinity means is the invitation contained in this book. And, more than that, in these pages is an invitation to engage in mathematics as a creative, collaborative endeavour, linked to our intuitions and emotions and of relevance to our lives. This Second Edition, coming nearly 30 years after the first, takes classic ideas and brings them up to date with the smartphone age making it fascinating and relevant to anyone loosely interested in popular science, or more squarely, with the Queen of Sciences itself, Mathematics.