I read the following statement in one of Penrose's paper
zero rest-mass field equations can, with suitable interpretations, be regarded as being conformally invariant.
I take this to imply that if I would like to describe massless scalar fields (for example) in curved spacetimes I should couple them conformally. More precisely, the curved space generalization of the action is $$ - \phi \partial^2 \phi \to - \phi \left( \nabla^2 + \frac{1}{6} R \right) \phi ~~~(d=4) $$ instead of the naive $\partial^2 \to \nabla^2$. Why is this the case?
EDIT: I believe this holds for massive scalar fields as well, though we no longer have conformal invariance.