If the CDM paradigm is correct, then dwarf galaxies are far rarer than dark matter haloes of comparable circular velocity at the present day. Some process or processes must suppress galaxy formation almost completely on these small scales. Several plausible mechanisms may explain the scarcity of dwarf galaxies in the local universe, but it has been hard to distinguish between the different models observationally. The spatial distribution and kinematics of dwarf galaxies contain additional information about the age and formation history of these systems. Interpreted in the CDM framework, they provide powerful new constraints on the physics of dwarf suppression, ruling out at high significance the simplest models, in which the halo occupation probability is constant above some fixed mass or velocity threshold. We describe recent tests of small-scale galaxy formation models using the satellites of the local group and other nearby groups, and discuss the implications of these results for studies of the stellar halo, sub-galactic dark matter and the reionization of the universe at high redshift.