  The running coupling constants (in particular, the gravitational one) are
studied in asymptotically free GUTs and in finite GUTs in curved spacetime,
with explicit examples. The running gravitational coupling is used to calculate
the leading quantum GUT corrections to the Newtonian potential, which turn out
to be of logarithmic form in asymptotically free GUTs. A comparison with the
effective theory for the conformal factor ---where leading quantum corrections
to the Newtonian potential are again logarithmic--- is made. A totally
asymptotically free $O(N)$ GUT with quantum higher derivative gravity is then
constructed, using the technique of introducing renormalization group (RG)
potentials in the space of couplings. RG equations for the cosmological and
gravitational couplings in this theory are derived, and solved numerically,
showing the influence of higher-derivative quantum gravity on the Newtonian
potential. The RG-improved effective gravitational Lagrangian for
asymptotically free massive GUTs is calculated in the strong (almost constant)
curvature regime, and the non-singular De Sitter solution to the quantum
corrected gravitational equations is subsequently discussed. Finally, possible
extensions of the results here obtained are briefly outlined.
