Deviations from the $1/r^{2}$ Newton law due to extra dimensions and $\alpha'$-corrections

We systematically examine corrections to the gravitational inverse square law, which are due to compactified extra dimensions, and to higher-derivative R^2-terms in the effective gravitational action. We find that both induce Yukawa-type potentials for which we calculate the strength \alpha and range. In general the range of the Yukawa correction is given by the wavelength of the lightest Kaluza-Klein state and its strength, relative to the standard gravitational potential, by the corresponding degeneracy. In particular, when n extra dimensions are compactified on an n-torus, we find that the strength of the potential is \alpha=2n, whereas the compactification on an n-sphere gives are considered, we find a repulsive Yukawa potential with \alpha=-1.