Maryna Viazovska recently made a stunning breakthrough on sphere packing by showing the E8 root lattice gives the densest packing of spheres in 8 dimensional space [arxiv:1603.04246]. This is the first result of its kind for dimensions $> 3$, and follows an approach suggested by Cohn-Elkies from 1999 via harmonic analysis. The talk will describe this method, her proof, as well as new joint work with Viazovska, Cohn, Kumar, and Radchenko that completely solves the sphere packing problem in dimension 24 [arxiv:1603.06518]: the Leech lattice gives the densest packing of spheres in 24 dimensions, and no other periodic packing is as dense as it.