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.
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