  The cold dark matter (CDM) cosmological model has been remarkably successful
in explaining cosmic structure over an enormous span of redshift, but it has
faced persistent challenges from observations that probe the innermost regions
of dark matter halos and the properties of the Milky Way's dwarf galaxy
satellites. We review the current observational and theoretical status of these
"small scale controversies." Cosmological simulations that incorporate only
gravity and collisionless CDM predict halos with abundant substructure and
central densities that are too high to match constraints from galaxy dynamics.
The solution could lie in baryonic physics: recent numerical simulations and
analytic models suggest that gravitational potential fluctuations tied to
efficient supernova feedback can flatten the central cusps of halos in massive
galaxies, and a combination of feedback and low star-formation efficiency could
explain why most of the dark matter subhalos orbiting the Milky Way do not host
visible galaxies. However, it is not clear that this solution can work in the
lowest mass galaxies where discrepancies are observed. Alternatively, the
small-scale conflicts could be evidence of more complex physics in the dark
sector itself. For example, elastic scattering from strong dark matter
self-interactions can alter predicted halo mass profiles, leading to good
agreement with observations across a wide range of galaxy mass. Gravitational
lensing and dynamical perturbations of tidal streams in the stellar halo
provide evidence for an abundant population of low mass subhalos in accord with
CDM predictions. These observational approaches will get more powerful over the
next few years.
