This book develops the theory of partitions. Simply put, the partitions of a number are the ways of writing that number as sums of positive integers. For example, the five partitions of 4 are 4: 3+1, 2+2, 2+1+1, and 1+1+1+1. Surprisingly, such a simple matter requires some deep mathematics for its study. This book considers the many theoretical aspects of this subject, which have in turn recently found applications to statistical mechanics, computer science and other branches of mathematics. With minimal prerequisites, this book is suitable for students as well as researchers in combinatorics, analysis, and number theory.