In this paper we establish eigenvector delocalization and bulk universality for Lévy matrices, which are real, symmetric, $N \times N$ random matrices $\textbf{H}$ whose upper triangular entries are independent, identically distributed $α$-stable laws. First, if $α\in (1, 2)$ 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 $α\in (0, 2)$, there exists a constant $c(α) > 0$ such that the same statements hold if $|E| < c (α)$.