Galaxies constitute a powerful probe of the LCDM cosmological model at the smallest nonlinear scales, where no other robust observational probes are yet available. Using the number density of galaxies as a function of their rotation velocity (the galaxy velocity function, VF) we are able to confront theoretical predictions with observations of the nearby universe. Using the most sensitive currently available data we find a significant underabundance of structure at scales below the mass of the Small Magellanic Cloud. A complete accounting for baryonic effects including stellar feedback and UV photoheating does not resolve the discrepancy. After exploring a variety of solutions including warm or self-interacting dark matter (DM) as well as cosmological parameter variations we find that while the low-mass end of the VF is sensitive to DM particle physics, the massive end is a sensitive probe of cosmology. Recent hydrodynamic galaxy simulations offer solutions to the problem of the "missing dwarfs" but their detailed kinematics fail to reproduce those of the faintest observed galaxies. With the advent of the unprecedented HI surveys with the SKA and its precursors in the next few years, “observing” tightly constrained simulations will be key to understanding the influence of galaxy formation physics on the recovery of the potential from dwarf galaxy rotation curves and on their use as near-field cosmological probes.