  We study both analytically and numerically the gravitational fields of stars
in f(R) gravity theories. We derive the generalized Tolman-Oppenheimer-Volkov
equations for these theories and show that in metric f(R) models the
Parameterized Post-Newtonian parameter $\gamma_{\rm PPN} = 1/2$ is a robust
outcome for a large class of boundary conditions set at the center of the star.
This result is also unchanged by introduction of dark matter in the Solar
System. We find also a class of solutions with $\gamma_{\rm PPN} \approx 1$ in
the metric $f(R)=R-\mu^4/R$ model, but these solutions turn out to be unstable
and decay in time. On the other hand, the Palatini version of the theory is
found to satisfy the Solar System constraints. We also consider compact stars
in the Palatini formalism, and show that these models are not inconsistent with
polytropic equations of state. Finally, we comment on the equivalence between
f(R) gravity and scalar-tensor theories and show that many interesting Palatini
f(R) gravity models can not be understood as a limiting case of a
Jordan-Brans-Dicke theory with $\omega \to -3/2$.
