  We study conformal mappings in the Grushin plane and provide a number of
their characterizations in terms of the Sobolev mappings and their geometry.
Furthermore, we connect conformality on the Grushin plane with conformality on
the complex plane by using the Meyerson map. Among applications we discuss
admissible curves and length-distortion estimates in the Grushin plane, as well
as the Carath\'eodory extension theorem.
