  We consider the ensemble of adjacency matrices of Erd{\H o}s-R\'enyi random
graphs, i.e.\ graphs on $N$ vertices where every edge is chosen independently
and with probability $p \equiv p(N)$. We rescale the matrix so that its bulk
eigenvalues are of order one. Under the assumption $p N \gg N^{2/3}$, we prove
the universality of eigenvalue distributions both in the bulk and at the edge
of the spectrum. More precisely, we prove (1) that the eigenvalue spacing of
the Erd{\H o}s-R\'enyi graph in the bulk of the spectrum has the same
distribution as that of the Gaussian orthogonal ensemble; and (2) that the
second largest eigenvalue of the Erd{\H o}s-R\'enyi graph has the same
distribution as the largest eigenvalue of the Gaussian orthogonal ensemble. As
an application of our method, we prove the bulk universality of generalized
Wigner matrices under the assumption that the matrix entries have at least $4 +
\epsilon$ moments.
