Lévy matrices are symmetric random matrices whose entries are independent alpha-stable laws. Such distributions have infinite variance, and when alpha is less than 1, infinite mean. In the latter case these matrices are conjectured to exhibit a sharp transition from a delocalized regime at low energy to a localized regime at high energy, like the infamous Anderson model in mathematical physics. We discuss work establishing the existence of a delocalized regime with GOE eigenvalue statistics. Further, we characterize the eigenvector statistics in this regime and find they display novel, non-Gaussian behavior. This talk describes joint works with Amol Aggarwal, Jake Marcinek, and Horng-Tzer Yau.