  In this paper we establish eigenvector delocalization and bulk universality
for L\'{e}vy matrices, which are real, symmetric, $N \times N$ random matrices
$\textbf{H}$ whose upper triangular entries are independent, identically
distributed $\alpha$-stable laws. First, if $\alpha \in (1, 2)$ and $E \in
\mathbb{R}$ is any energy bounded away from $0$, we show that every eigenvector
of $\textbf{H}$ corresponding to an eigenvalue near $E$ is completely
delocalized and that the local spectral statistics of $\textbf{H}$ around $E$
converge to those of the Gaussian Orthogonal Ensemble (GOE) as $N$ tends to
$\infty$. Second, we show for almost all $\alpha \in (0, 2)$, there exists a
constant $c(\alpha) > 0$ such that the same statements hold if $|E| < c
(\alpha)$.
