One of the distinctive feature of the $d$-dimensional torus $T^d$ is that it admits a faithful smooth action by $\mathrm{SL}_d(\mathbb Z)$, so one might wonder whether such an action (or any nontrivial action) also exists for exotic tori i.e. smooth $d$-manifolds that are homeomorphic but not diffeomorphic to $T^d$. I will discuss this and related questions in the talk, based on joint work with M. Bustamante, A. Kupers, and B. Tshishiku.