What is the densest possible packing of identical spheres in Euclidean space? The answer to this very old question in mathematics is only known in dimensions 1, 2, 3, 8, and 24. In very high dimensions, the answer is almost a complete mystery. I will describe a connection between geometric packing problems such as this and a central model in statistical physics: the hard sphere model in which gas molecules are represented by non-overlapping spheres. I will then present a new lower bound on kissing numbers in high dimensions.