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In 1998 Thomas Hales dramatically announced the solution to the problem which has long teased eminent mathematicians: what is the densest possible arrangement of identical spheres? This text recounts the story of this problem and many others which have to do with packing things together. The examples range through mathematics, physics, biology and engineering; they include the arrangement of soap bubbles in a foam, atoms in a crystal, the architecture of the bee's honeycomb, and the structure of the Giant's Causeway. While conveying all of the essential information in each case, with key references, the style is informal. Included are brief accounts of the lives of many of the scientists who devoted themselves to problems of packing over many centuries, together with numerous comments on their efforts.