  Ahlfors and Gehring asked for the Riemann Mapping Theorem for quasiconformal
mappings (QC) of R^3. We summarise our solution: (a) QC reflections are tame
(b) T is the fixed set of a QC reflection iff T is a uniform sphere (i.e. the
limits of its blowups are topologically flat spheres) (c) T is a quasiphere iff
is a uniform & quasisymmetric sphere (d) A domain D is the QC image of the unit
ball iff it is a regular ball: uniformly simply connected (i.e. the limits of
blowups are topological balls) with boundary mapping from the unit sphere which
is regular (i.e. the limits of blowups are boundary maps from the sphere). The
latter is equivalent to being locally linearly connected and Lowner.
