Infinity is a fascinating subject. What do we mean when we use the concept of infinity? What did the ancient Greeks mean? What do modern mathematicians mean when they use infinity in their calculations? What are the debates of contemporary philosophers of mathematics regarding the infinite? This book, starting from Aristotle and Plato, tries to retrace the steps that led to contemporary conceptions of infinity. The book also deals with the first mathematical problem of the twentieth century on Hilbert's list with no solution that has correlations with the concept of infinity. Is the mathematical universe unique or do we have infinite parallel universes with different truths? This is a question related to the concept of infinity and the book tries to answer it. The goal of this book is to make the mathematics of infinity accessible by showing the most incredible mathematical results of the last century and deepening the themes of contemporary debates in mathematics and philosophy regarding infinity. EMANUELE GAMBETTA obtained his PHD in philosophy of mathematics from the Scuola Normale Superiore with cum laude mark. He was a visiting research student at the university of Barcelona where he participated in the seminars of the set theory center. He has been a visiting graduate student in many universities around the world.