  The notion of Schwarzian derivative for locally univalent holomorphic
functions on complex plane was generalized for conformal diffeomorphisms by
Osgood and Stowe in 1992 [27]. We shall identify a tensor that may serve as an
analogue of the Schwarzian of Osgood and Stowe for CR mappings, and then use
the tensor to define and study the CR M\"obius transformations and metrics of
pseudo-hermitian manifolds. We shall establish basic properties of the CR
Schwarzian and a local characterization of CR spherical manifolds in terms of
fully integrability of the CR M\"obius equation. In another direction, we shall
prove two rigidity results for M\"obius change of metrics on compact CR
manifolds.
