  We address the galaxy rotation curves through the Yukawa gravitational
potential emerging as a correction of the Newtonian potential in extended
theories of gravity. On the one hand, we consider the contribution of the
galactic bulge, galactic disk, and the dark matter halo of the
Navarro-Frenk-White profile, in the framework of the standard $\Lambda$CDM
model. On the other hand, we use modified Yukawa gravity to show that the
rotational velocity of galaxies can be addressed successfully without the need
for dark matter. In Yukawa gravity, we recover MOND and show that dark matter
might be seen as an apparent effect due to the modification of the law of
gravitation in terms of two parameters: the coupling constant $\alpha $ and the
characteristic length $\lambda$. We thus test our theoretical scenario using
the Milky Way and M31 rotation velocity curves. In particular, we place
observational constraints on the free parameters of Yukawa cosmology through
the Monte Carlo method and then compare our results with the predictions of the
$\Lambda$CDM paradigm by making use of Bayesian information criteria.
Specifically, we find that $\lambda$ is constrained to be of the order of kpc,
while cosmological data suggest $\lambda$ of the order of Gpc. To explain this
discrepancy, we argue that there is a fundamental limitation in measuring
$\lambda$ due to the role of quantum mechanics on cosmological scales.
