Fourth Order Theories of Gravity have recently attracted a lot of interest as candidates to explain the observed cosmic acceleration, the flatness of the rotation curves of spiral galaxies, the large scale structure and other relevant astrophysical phenomena. This means that the ”Dark Side" issue of the Universe could be completely reversed considering dark matter and dark energy as ”shortcomings” of General Relativity in its simplest formulation (a linear theory in the Ricci scalar R, minimally coupled to the standard perfect fluid matter) and claiming for the ”correct” theory of Gravity as that derived by matching the largest number of observational data, without imposing any theory a priori. As a working hypothesis, accelerating behavior of cosmic fluid, large scale structure, potential of galaxy clusters, rotation curves of spiral galaxies could be reproduced by means of extending General Relativity to generic actions containing higher order and non-minimally coupled terms in curvature invariants. In other words, gravity could act in different ways at different scales and the above "shortcomings" could be due to the incorrect extrapolations of the Einstein theory, actually tested at short scales and low energy regimes. Very likely, what we call ”dark matter and "dark energy" could be nothing else but signals of the breakdown of General Relativity at large scales. Then, it is a crucial point testing these Extended Theories in the weak field limit. In this sense, comparing these theories to General Relativity could be a fundamental step to retain or rule out them. In this review paper, after a survey of what is intended for Extended Theories of Gravity in the so called metric approach,we extensively discuss their Newtonian and the post-Newtonian limits pointing out, in details, their resemblances and differences with respect to General Relativity. Particular emphasis is placed on the exact solutions and methods used to obtain them. Finally,it is clearly shown that General Relativity results, in the Solar System context, are easily recovered since Einstein theory is a particular case of this extended approach. This is a crucial point against several wrong results in literature stating that these theories (e.g. f(R)-gravity) are not viable at local scales.
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