The problem of finding the densest way to pack equally sized spheres in Euclidean space has been studied for hundreds of years. The densest packing is only known in dimensions 1, 2, 3, 8, and 24, with the last two being recently solved by Viazovska (2016) and Cohn et al. (2016). This talk will give an overview of several lower and upper bounds for the maximum packing density. We will also show examples of dense sphere packings in small and high dimensions and explain their properties and connections to other areas of mathematics.