We consider the family or degree two rational maps of the Riemann sphere to itself that preserve the real line, with a goal of understanding the parameter space of all such mappings.  As in the case of polynomial maps,  the postcritically finite case (which are Thurston maps) play an important role; we also discuss the associated combinatorics of such maps.  [Joint work with Araceli Bonifant and John Milnor.]

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