  We investigate the astrophysical and cosmological implications of the
recently proposed idea of a running gravitational coupling on macroscopic
scales. We find that when applied to the rotation curves of galaxies, their
flatness requires still the presence of dark matter. Bounds on the variation of
the gravitational coupling from primordial nucleosynthesis, change of the
period of binary pulsars, gravitational lensing and the cosmic virial theorem
are analysed.
