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.