  We extend the f(R) gravity action by including a generic dependence upon the
Weyl tensor, and further generalize it to supergravity by using the
super-curvature (R) and super-Weyl (W) chiral superfields in N=1 chiral curved
superspace. We argue that our (super)gravitational actions are the meaningful
extensions of the phenomenological f(R) gravity and its locally supersymmetric
generalization towards their UV completion and their embedding into superstring
theories. The proposed actions can be used for study of cosmological
perturbations and gravitational instabilities due to a nonvanishing Weyl tensor
in gravity and supergravity.
