We discuss recent work proving eigenvector delocalization and bulk eigenvalue universality for symmetric random matrices whose entries are independent \alpha-stable laws with \alpha <2. Such distributions have infinite variance, and when \alpha <1, infinite mean. In the latter case such 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. These results are joint work with Amol Aggarwal and Horng-Tzer Yau.