We consider Hermitian and symmetric random band matrices on the $d$-dimensional lattice $(\mathbb{Z}/L\mathbb{Z})^d$ with bandwidth $W$, focusing on local eigenvalue statistics at the spectral edge in the limit $W\to\infty$. Our analysis reveals a critical dimension $d_c=6$ and identifies the critical bandwidth scaling as $W_c=L^{(1-d/6)_+}$. In the Hermitian case, we establish the Anderson transition for all dimensions $d<4$, and GUE edge universality when $d\geq 4$ under the condition $W\geq L^{1/3+ε}$ for any $ε>0$. In the symmetric case, we also establish parallel but more subtle transition phenomena after tadpole diagram renormalization. These findings extend Sodin's pioneering work [Ann. Math. 172, 2010], which was limited to the one-dimensional case and did not address the critical phenomena.