In the 1990's Metsch showed that the Grassmann graphs $J_q(n, D)$ are characterized by their intersection numbers if $n \geq \max\{2D+2, 2D+6-q\}$. In 2005 Van Dam and Koolen found the twisted Grassmann graphs $\tilde{J}_q(2D+1, D)$ with the same intersection numbers as $J_q(2D+1, D)$.  In this talk I will discuss the Grassmann graphs $J_q(2D, D)$. This is based on joint work with A. Gavrilyuk (PNU).