  The cold dark matter model successfully explains both the emergence and
evolution of cosmic structures on large scales and, when we include a
cosmological constant, the properties of the homogeneous and isotropic
Universe. However, the cold dark matter model faces persistent challenges on
the scales of galaxies. {Indeed,} N-body simulations predict some galaxy
properties that are at odds with the observations. These discrepancies are
primarily related to the dark matter distribution in the innermost regions of
the halos of galaxies and to the dynamical properties of dwarf galaxies. They
may have three different origins: (1) the baryonic physics affecting galaxy
formation is still poorly understood and it is thus not properly included in
the model; (2) the actual properties of dark matter differs from those of the
conventional cold dark matter; (3) the theory of gravity departs from General
Relativity. Solving these discrepancies is a rapidly evolving research field.
We illustrate some of the solutions proposed} within the cold dark matter
model, and solutions when including warm dark matter, self-interacting dark
matter, axion-like particles, or fuzzy dark matter. { We also illustrate some
modifications of the theory of gravity: Modified Newtonian Dynamics (MOND),
MOdified Gravity (MOG), and $f(R)$ gravity.
