  The amount of dark matter in the Milky Way and beyond is examined by taking
into account the possible running of the gravitational constant $G$ as a
function of distance scale. If the running of $G$, as suggested by the
Asymptotically-Free Higher-Derivative quantum gravity, is incorporated into the
calculation of the total dark matter in the galactic halo, the amount of dark
matter that is necessary to explain the rotation curve is shown to be reduced
by one third compared with the standard calculations. However, this running of
$G$ alone cannot reproduce the observed flat behavior of the rotation curve. It
is also shown that the running of $G$ cannot explain away the presence of most
of the dark matter beyond the scale of $ \sim 10$ Mpc in the Universe. We also
present a pedagogical explanation for the running of $G(r)$ in the region of
large scales which is clearly a classical domain.
