The quantum effects of matter fields on a classical curved
background may lead to the phenomenologically interesting
contributions to the action of gravity. The present day
knowledge of quantum field theory in curved space enables
one to calculate some part of such corrections, mainly the
ones in the higher derivative sector of the theory. At the
same time, the derivation of the most interesting corrections
to the cosmological constant and Einstein-Hilbert terms are
beyond our possibilities, such that one has to rely, at least
in part, to the phenomenological approach. In the cosmological
setting the form of quantum contributions can be established
from the covariance arguments. As a result we arrive at the
form of the possible quantum terms which depend on a
single free parameter. This parameter can be indeed restricted,
e.g., from the analysis of the LSS data. In the astrophysical
case we meet a bit more complicated situation, and the most
challenging problem is to find an appropriate physical
identification for the scale parameter of the renormalization
group. It is remarkable that one of the natural choices of this
identification provides an extremely good fit for the rotation
curves for the sample of distinct galaxies without invoking the
CDM concept.