We will discuss what a (lattice) sphere packing is and how its density can be defined. Then, we will look at examples in dimensions 2, 3, and perhaps 4. We will also introduce the related kissing number problem. Finally, we will survey known results and look at a rather puzzling plot of the densities — by dimension — of the densest sphere packings known. If we have time, we can discuss the two lattices (E_8 and Leech lattice) that lead to the packings in dimensions 8 and 24 that were recently shown to have highest possible density.