  A new class of conformal field theories is presented, where the background
gravitational field is conformally flat. Conformally flat (CF) spacetimes enjoy
conformal properties quite similar to the ones of flat spacetime. The conformal
isometry group is of maximal rank and the conformal Killing vectors in
conformally flat coordinates are {\em exactly} the same as the ones of flat
spacetime. In this work, a new concept of distance is introduced, the {\em
conformal distance}, which transforms covariantly under all conformal
isometries of the CF space. It is shown that precisely for CF spacetimes, an
adequate power of the said conformal distance is a solution of the non-minimal
d'Alembert equation.
