The sphere packing problem in dimension $n$ asks:
How densely can one pack identical Euclidean balls in $\mathbb{R}^n$ with
disjoint interiors? We review some of this problem's history and
connections with various areas of mathematics and science.
Some special values of $n$, notably $8$ and $24$, allow for remarkably
tight and symmetrical configurations that have long been suspected
to be the densest possible in those dimensions. We conclude with the
series of recent results culminating with Viazovska's breakthrough that
led to the solution of the sphere packing problem for $n=8$ and $n=24$.