A long-standing problem in metric geometry is to recognize those metric spaces that admit a “good” parametrization by the Euclidean space or the unit sphere. In this talk, we will discuss the quasisymmetric uniformization problem and some recent advancements in terms of the Menger curvature, a curvature suited for Euclidean sets with no a priori smooth structure. The talk is based on a joint work with G. C. David (Ball State University).