We define the moments of a random finite abelian group, and discuss the moment problem, i.e. when a random finite abelian group is determined by its moments. This will play an important role in our discussion of a universality theorem, an analog for random groups of the central limit theorem, in which finite abelian groups built from generators and independent random relations converge to a universal random group not depending on the distribution from which the relations are drawn. We discuss applications of these ideas, including to the distribution of sandpile groups of random graphs, Mészáros's result on the non-singularity of the adjacency matrices of random d-regular graphs, and the probability that a random 0/1 matrix gives a surjective map on integral vectors. The background assumed in the talk will be minimal (finite abelian groups, basic definitions in probability). This talk will include joint with with Hoi Nguyen.