  The Higher Order Theories of Gravity - $f(R, R_{\alpha\beta}R^{\alpha\beta})$
- theory, where $R$ is the Ricci scalar, $R_{\alpha\beta}$ is the Ricci tensor
and $f$ is any analytic function - 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 and other relevant
astrophysical phenomena. It is a crucial point testing these alternative
theories in the so called weak field and newtonian limit of a $f(R,
R_{\alpha\beta}R^{\alpha\beta})$ - theory. With this "perturbation technique"
it is possible to find spherically symmetric solutions and compare them with
the ones of General Relativity. On both approaches we found a modification of
General Relativity: the behaviour of gravitational potential presents a
modification Yukawa - like in the newtonian case and a massive propagation in
the weak field case. When the modification of the theory is removed (i.e. $f(R,
R_{\alpha\beta}R^{\alpha\beta}) = R$, Hilbert - Einstein lagrangian) we find
the usual outcomes of General Relativity. Also the Noether symmetries technique
has been investigated to find some time independent spherically symmetric
solutions.
