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 analytical 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.dark matter | cosmology | galaxy formation T he cold dark matter (CDM) hypothesis-that dark matter consists of a weakly interacting particle whose velocity dispersion in the early universe was too small to erase structure on a galactic or subgalactic scale-emerged in the early 1980s and quickly became a central element of the theory of cosmic structure formation. Influential early papers include Peebles' calculation of cosmic microwave background (CMB) anisotropies and the matter power spectrum (1), discussions of galaxy formation with particle dark matter by Bond et al. (2) and Blumenthal et al. (3,4), and Davis et al.'s (5) numerical simulations of galaxy clustering. By the mid-1990s, the simplest CDM model with scale-invariant primordial fluctuations and a critical matter density of ðΩ m = 1Þ had run afoul of multiple lines of observational evidence, including the shape of the galaxy power spectrum, estimates of the mean matter density from galaxy clusters and galaxy motions, the age of the universe inferred from estimates of the Hubble constant, and the amplitude of matter clustering extrapolated forward from the fluctuations measured in the CMB. Many variants on "canonical" CDM were proposed to address these challenges, and by the turn of the century, the combination of supernova evidence for cosmic acceleration and CMB evidence for ...