  We discuss the nature of quantum field theories involving gravity that are
classically scale-invariant. We show that gravitational radiative corrections
are crucial in the determination of the nature of the vacuum state in such
theories, which are renormalisable, technically natural, and can be
asymptotically free in all dimensionless couplings. In the pure gravity case,
we discuss the role of the Gauss-Bonnet term, and we find that Dimensional
Transmutation (DT) \`a la Coleman-Weinberg leads to extrema of the effective
action corresponding to nonzero values of the curvature, but such that these
extrema are local maxima. In even the simplest extension of the theory to
include scalar fields, we show that the same phenomenon can lead to extrema
that are local minima of the effective action, with both non-zero curvature and
non-zero scalar vacuum expectation values, leading to spontaneous generation of
the Planck mass. Although we find an asymptotically free (AF) fixed point
exists, unfortunately, no running of the couplings connect the region of DT to
the basin of attraction of the AF fixed point. We also find there remains a
flat direction for one of the conformal modes. We suggest that in more
realistic models AF and DT could be compatible, and that the same scalar vacuum
expectation values could be responsible both for DT and for spontaneous
breaking of a Grand Unified gauge group.
