The sphere packing problem asks for the maximum proportion of d-dimensional Euclidean space that can be covered by disjoint identical spheres.  The first result we will discuss is a new lower bound for this problem, which is the first asymptotically growing improvement in all high dimensions since Rogers' bound from 1947.  We will then discuss connections to statistical physics and describe problems concerning the structure of random sphere packings in both Euclidean and non-Euclidean spaces.