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Higher order theories of gravity,
which can be related to several quantum gravity approaches,
have recently attracted a lot of interest as alternative candidates to explain the observed cosmic acceleration,
the flatness of the rotation curves of spiral galaxies,
the gravitational potential of galaxy clusters,
and other relevant astrophysical phenomena.
Very likely,
what we call ''dark matter'' and ''dark energy'' are nothing else but signals of the breakdown of General Relativity at large scales.
Furthermore,
PPN-parameters deduced from Solar System experiments do not exclude,
a priori,
the possibility that such theories could give small observable effects also at these scales.
We review these results giving the basic ingredients of such an approach.
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