We consider a general class of random band matrices H on the d-dimensional lattice of linear size L. Its entries $$h_{xy}$$ are independent random variables up to symmetry $$H=H^*$$ and are negligible if $$|x-y|$$ exceeds the band width W. It is conjectured that a sharp localization-delocalization transition of bulk eigenvectors occurs at a critical band width $$W=L^a$$, and the critical exponent is $$a=\frac{1}{2}$$ in dimension 1 and $$a=0$$ in dimensions 2 and higher. In this talk, I will discuss about some recent progresses about the delocalized phase of random band matrices. In particular, we show that the critical exponents is at most $$\frac{3}{4}$$ in dimension 1, at most $$\frac{2}{d+2}$$ in dimensions 3 to 7, and equal to 0 in dimensions 8 and higher. Based on joint work with Bourgade, Yau and Yin.