Consider an $ N \times N$ Hermitian one-dimensional random band matrix with  band width $W > N^{1 / 2 + \varepsilon} $ for any $ \varepsilon > 0$. In a joint work with J. Yin, we proved that all  eigenvectors  are delocalized in high probability and  universality of local eigenvalue statistics holds  in the bulk of the spectrum  in the large $N$ limit.  These results were extended to dimension $d=2$ in a joint work with S. Dubova, K. Yang and J. Yin.