  General Relativity is able to describe the dynamics of galaxies and larger
cosmic structures only if most of the matter in the Universe is dark, namely it
does not emit any electromagnetic radiation. Intriguingly, on the scale of
galaxies, there is strong observational evidence that the presence of dark
matter appears to be necessary only when the gravitational field inferred from
the distribution of the luminous matter falls below an acceleration of the
order of 10^(-10) m/s^2. In the standard model, which combines Newtonian
gravity with dark matter, the origin of this acceleration scale is challenging
and remains unsolved. On the contrary, the full set of observations can be
neatly described, and were partly predicted, by a modification of Newtonian
dynamics, dubbed MOND, that does not resort to the existence of dark matter. On
the scale of galaxy clusters and beyond, however, MOND is not as successful as
on the scale of galaxies, and the existence of some dark matter appears
unavoidable. A model combining MOND with hot dark matter made of sterile
neutrinos seems to be able to describe most of the astrophysical phenomenology,
from the power spectrum of the cosmic microwave background anisotropies to the
dynamics of dwarf galaxies. Whether there exists a yet unknown covariant theory
that contains General Relativity and Newtonian gravity in the weak field limit,
and MOND as the ultra-weak field limit is still an open question.
