  A general analytic procedure is developed to deal with the Newtonian limit of
$f(R)$ gravity. A discussion comparing the Newtonian and the post-Newtonian
limit of these models is proposed in order to point out the differences between
the two approaches. We calculate the post-Newtonian parameters of such theories
without any redefinition of the degrees of freedom, in particular, without
adopting some scalar fields and without any change from Jordan to Einstein
frame. Considering the Taylor expansion of a generic $f(R)$ theory, it is
possible to obtain general solutions in term of the metric coefficients up to
the third order of approximation. In particular, the solution relative to the
$g_{tt}$ component gives a gravitational potential always corrected with
respect to the Newtonian one of the linear theory $f(R)=R$. Furthermore, we
show that the Birkhoff theorem is not a general result for $f(R)$-gravity since
time-dependent evolution for spherically symmetric solutions can be achieved
depending on the order of perturbations. Finally, we discuss the
post-Minkowskian limit and the emergence of massive gravitational wave
solutions.
