  We study the renormalization group flow in a class of scalar-tensor theories
involving at most two derivatives of the fields. We show in general that
minimal coupling is self consistent, in the sense that when the scalar self
couplings are switched off, their beta functions also vanish. Complete,
explicit beta functions that could be applied to a variety of cosmological
models are given in a five parameter truncation of the theory in $d=4$. In any
dimension $d>2$ we find that the flow has only a "Gaussian Matter" fixed point,
where all scalar self interactions vanish but Newton's constant and the
cosmological constant are nontrivial. The properties of these fixed points can
be studied algebraically to some extent. In $d=3$ we also find a
gravitationally dressed version of the Wilson-Fisher fixed point, but it seems
to have unphysical properties. These findings are in accordance with the
hypothesis that these theories are asymptotically safe.
