There is an essentially complete renormalization theory for analytic critical circle maps which serves to justify the golden mean universality and other deep conjectures by physicists. Its development ran parallel with the universality of Feigenbaum constants in the framework of unimodal and quadratic-like maps. In this talk, I will introduce a generalization called "critical quasicircle maps", i.e. analytic self homeomorphisms of a quasicircle with a single critical point. We will sketch the realization of such maps with any prescribed combinatorics not necessarily equivalent to critical circle maps. We will then discuss the state of the art renormalization picture together with its implications on rigidity, universality, and regularity of conjugacy classes.