The introduction of quasiconformal surgery in dynamics by Sullivan was crucial in settling several conjectures, but it was soon realized that the method has its limitations. In fact, in many problems the underlying conformal structures have unbounded dilatations and quasiconformal machinery fails. Trans-quasiconformal maps believed to be the right tool for these problems, but it took a long time until they could besuccessfully applied in this dynamical context. In this talk, I will give an expository account of some of these developments, especially my joint work with Petersen. I will also discuss some geometric/analytic properties of trans-quasiconformal maps, including how they distort Hausdorff dimension.''