L\'{e}vy matrices are symmetric matrices whose entries are random variables with infinite variance; they are governed by a parameter $\alpha \in (0, 2)$ dictating the power law decay of their entries. For $\alpha <1$, they are believed to serve as one of the few examples of a matrix model exhibiting a mobility edge, also called an Anderson transition, that separates chaotic (GOE) eigenvalue spacing statistics from ordered (Poisson) ones. In this talk we describe results concerning the statistics for the eigenvalue spacings and eigenvector entries of L\'{e}vy matrices. In particular, for $\alpha \in (1, 2)$ their eigenvalue statistics asymptotically follow those of the GOE throughout the spectrum, and for $\alpha < 1$ the same statement holds around small eigenvalues. \\ \\ These describe joint works with Patrick Lopatto, Jake Marcinek, and Horng-Tzer Yau.