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The goal of this survey is to present intimate interactions between fourbranches of conformal dynamics: iterations of anti-rational maps, actions ofKleinian reflection groups, dynamics generated by Schwarz reflections inquadrature domains, and algebraic correspondences. We start with severalexamples of Schwarz reflections as well as algebraic correspondences obtainedby matings between anti-rational maps and reflection groups, and examples ofJulia set realizations for limit sets of reflection groups (including classicalApollonian-like gaskets). We follow up these examples with dynamical relationsbetween explicit Schwarz reflection parameter spaces and parameter spaces ofanti-rational maps and of reflection groups. These are complemented by a numberof general results and illustrations of important technical tools, such asDavid surgery and straightening techniques. We also collect several analyticapplications of the above theory.

Mirrors of conformal dynamics: Interplay between anti-rational maps, reflection groups, Schwarz reflections, and correspondences