Abstract We examine gravity theories derived from a gravitational lagrangian that is and analytic function ƒ(R) of the scalar curvature R in a space-time of arbitrary dimension D. We show that they are conformally equivalent to general relativity plus a scalar-field matter source with a particular self-interaction potential. The general form of this potential is calculated to be V(ø) = 1 2[Rƒ′(R)−ƒ(R)][ƒ′(R)] D (2−D) , where exp (ø) = [ƒ′ (R)] 2 (D−2) . Flat potentials arise as ø→+∞ for polynomial lagrangians of leading order β whenever D = 2 β. Several explicit examples are given and discussed with reference to inflation. We show how our results can lead to singularity theorems for gravity theories derived from a general lagrangian.