  For a general class of scalar--tensor gravity theories, we discuss how to
recover asymptotic freedom regimes when cosmic time $t\to\pm\infty$. Such a
feature means that the effective gravitational coupling $G_{eff}\to 0$, while
cosmological solutions can asymptotically assume de Sitter or power--law
behaviours. In our opinion, through this mechanism, it is possible to cure some
shortcomings in inflationary and in string--dilaton cosmology.
