In this lecture, we will review recent works regarding spectral statistics of the normalized adjacency matrices of random $d$-regular graphs on $N$ vertices.

Denote their eigenvalues by $\lambda_1=d/\sqrt{d-1}\geq \la_2\geq\la_3\cdots\geq \la_N$ and let $\gamma_i$ be the classical location of the $i$-th eigenvalue under the Kesten-McKay law. Our main result asserts that for any $d \ge 3$ the optimal eigenvalue rigidity holds in the sense that

\begin{align*}

|\lambda_i-\gamma_i|\leq \frac{N^{\oo_N(1)}}{N^{2/3} (\min\{i,N-i+1\})^{1/3}},\quad \forall i\in \{2,3,\cdots,N\}.

\end{align*}

with probability $1-N^{-1+\oo_N(1)}$. In particular, the characteristic $N^{-2/3}$ fluctuations for Tracy-Widom law is established for the second largest eigenvalue.

Furthermore, for $d \ge N^\varepsilon$ for any $\varepsilon > 0$ fixed, the extremal eigenvalues obey the Tracy-Widom law. This is a joint work with Jiaoyang Huang and Theo McKenzie.