Abstract: I will discuss recent results on describing the groups of quasisymmetric maps of the Julia sets of post-critically finite rational maps for different topological types: carpets, gaskets, and tree-like sets. I will mostly concentrate though on gasket Julia sets since there is no simple way to characterize gaskets topologically, while there is only one topological carpet and tree-like sets have rather simple characterization. In particular, I will discuss a construction of a large class of critically fixed (anti-)rational maps with gasket-like Julia sets. Such construction uses the Koebe--Andreev--Thurston Theorem on circle packings and the (anti-)rational maps are enumerated by topological triangulations of the sphere. The talk is based on joint work with Mario Bonk, Russell Lodge, Misha Lyubich, and Sabyasachi Mukherjee.