The conformal dimension of a metric space $(X,d)$, introduced by Pansu in 1989 in the context of Gromov-hyperbolic groups, is the infimum of the Hausdorff dimensions of all metric spaces quasisymmetrically equivalent to $(X,d)$. It is always bounded between the topological and Hausdorff dimensions of $(X,d)$. In this talk, we describe recent work showing that the conformal dimension of the Brownian sphere is exactly 2, matching its topological dimension. We also discuss a complementary result establishing that the Brownian sphere is quasisymmetrically rigid, meaning its group of quasisymmetric automorphisms is almost surely trivial. This talk is based on joint work with Yi Tian.