The aim of this chapter is to prove a marvellous theorem due to Hermann Minkowski in 1896. This asserts the existence, within a suitable set X , of a non-zero point of a lattice L, provided the volume of X is sufficiently large relative to that of a fundamental domain of L. The idea behind the proof is deceptive in its simplicity: X cannot be squashed into a space whose volume is less than that of X , unless X is allowed to overlap itself. Minkowski discovered that this essentially trivial observation has many non-trivial and important consequences, and used it as a foundation for an extensive theory of the ‘geometry of numbers’. As immediate and accessible instances of its application we prove the two-and four-squares theorems of classical number theory.