Lévy matrices are symmetric random matrices whose entry distributions have power law tails and infinite variance. They are predicted to exhibit an Anderson-type phase transition separating a region of delocalized eigenvectors from one with localized eigenvectors. We will discuss the context for this conjecture, and describe a result establishing it when the power law exponent is close to zero or one. This is joint work with Amol Aggarwal and Charles Bordenave.