A $7$-manifold with a $G_2$-structure $\varphi$ admits a $G_2$-connection with totally skew-symmetric torsion if and only if $d * \varphi = \theta \wedge * \varphi$, where $\theta$ is the Lee form of the $G_2$-structure. In the talk, I will present recent results on $7$-manifolds admitting a $G_2$-connection with closed totally skew-symmetric torsion. In particular, I will discuss the twisted $G_2$ equation, which represents the $G_2$-analogue of the twisted Calabi-Yau equation for $SU(n)$-structures introduced by Garcia-Fernández, Rubio, Shahbazi and Tipler. The talk is based on a joint work with Lucia Merchan and Alberto Raffero.