The best bounds known for the Euclidean sphere packing problem (at least in dimensions 4 through 36) are proved using linear programming bounds due to Cohn and Elkies. These bounds are derived using a simple argument based on Poisson summation, but optimizing the bounds has proved difficult. It is conjectured that the bounds are in fact sharp in 8 and 24 dimensions, and this has been verified to 30 decimal places, but it has not been proved. In this talk, I'll discuss the challenges that must be overcome to prove it as well as some new mysteries that have arisen recently in this area.