- We discuss ways of constructing theories of gravity where the action density is bounded from below. This property can make the euclidean path integral better defined than in the standard Einstein case. For a class of action densities which depend (non-linearly) on the scalar curvature, such models are at the classical level equivalent to scalar-tensor theories of gravity with scalar potentials that imply a stable vacuum (which can be flat space) and where there is the possibility of an inflationary phase, as well as other interesting features. Scalars with such types of potentials appear naturally in some Kaluza-Klein and other theories. When integrating out the scalars in such theories, we could end up with the models discussed here.