The classical sphere packing problem asks how densely one can pack balls with equal radii into R^n so that their interiors do not overlap and the lattice sphere packing problem is obtained by adding the additional hypothesis that the sphere centers form a lattice, i.e., an additive abelian group. These two problems have long and fascinating histories, spanning more than four centuries, and are currently solved only in dimensions up to 3 and in dimensions 1 through 8 and 24 respectively.

In this talk I will introduce both the sphere packing and lattice sphere packing problems, review some relevant background on lattices, and survey a few of the known results for these two problems. I will then describe my work with lattice sphere packings with extra algebraic structure. In particular, I will describe how lattices with extra algebraic structure can be used to prove an inequality bounding the densities of special classes of 2m and 4m-dimensional lattice sphere packings and how Hurwitz lattice sphere packings in quaternionic Hermitian space can be used to obtain new lower bounds for the sphere packing density in high dimensions divisible by four.