Sums of Squares of Integers is the first book to combine elementary methods with analytic methods of modular functions to study the representation of integers as sums of squares. It is the first to explain how to compute the number of representations of an integer n as the sum of s squares for any s and n. It is also the first to give a proof of Szemeredi's theorem, an important theorem in modern number theory, and it discusses how the modern theory of modular forms complements and clarifies the classical, fundamental results about sums of squares. The text includes applications to fields such as cryptography, crystallography, and thermal radiation.