The sphere packing problem asks to find the densest possible packing of identical spheres in d dimensions. The problem was recently solved analytically in 8 and 24 dimensions by Viazovska et al., building on linear programming bounds of Cohn+Elkies. I will show that there is a close connection between these results on sphere packing and the modular bootstrap in two-dimensional conformal field theories. In particular, I will explain that Viazovska's solution was essentially rediscovered in the conformal bootstrap literature in the guise of "analytic extremal functionals". It corresponds to saturation of the modular bootstrap bounds by known 2D CFTs. Sphere packing in a large number of dimensions maps to the modular bootstrap at large central charge, which can be used to constrain quantum gravity in large AdS_3. I will use the new analytic techniques to improve significantly on the best asymptotic upper bound on the mass of the lightest state in such theories.