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