There is a good deal of resemblence of CR geometry in dimension three with conformal geometry
in dimension four. Exploiting this resemblence is quite fruitful. For instance, the presence of
several conformally covariant operators in both geometries allows us to formulate correct
conditions for the embedding problem as well as the CR Yamabe problem. There is also
large difference in the presence of pluriharmonic functions. I will also describe a new
operator which gives control of the pluriharmonics and allows a formulation of a
sphere theorem in this geometry.