The problem of determining the local eigenvalue statistics (LES) for one-dimensional random band matrices (RBM) will be discussed with an emphasis on the localization regime. RBM are real symmetric $(2N+1) \times (2N+1)$ matrices with nonzero entries in a band of width $2 N^\alpha +1$ about the diagonal, for $0 \leq \alpha \leq 1.$ The nonzero entries are independent, identically distributed random variables. It is conjectured that as $N \rightarrow \infty$ and for $0 \leq \alpha '< \frac{1}{2}$, the LES is a Poisson point process, whereas for $\frac{1}{2} '< \alpha \leq 1$, the LES is the same as that for the Gaussian Orthogonal Ensemble. This corresponds to a phase transition from a localized to a delocalized state as $\alpha$ passes through $\frac{1}{2}$. In recent works with B. Brodie and with M. Krishna, we have made progress in proving this conjecture for $0 \leq \alpha '< \frac{1}{2}$.
Some of the results by others for the delocalized state with $\frac{1}{2} '< \alpha \leq 1$ will also be described.

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