  The observed number of dwarf galaxies as a function of rotation velocity is
significantly smaller than predicted by the standard model of cosmology. This
discrepancy cannot be simply solved by assuming strong baryonic feedback
processes, since they would violate the observed relation between maximum
circular velocity ($v_{\rm max}$) and baryon mass of galaxies. A speculative
but tantalising possibility is that the mismatch between observation and theory
points towards the existence of non-cold or non-collisionless dark matter (DM).
In this paper, we investigate the effects of warm, mixed (i.e warm plus cold),
and self-interacting DM scenarios on the abundance of dwarf galaxies and the
relation between observed HI line-width and maximum circular velocity. Both
effects have the potential to alleviate the apparent mismatch between the
observed and theoretical abundance of galaxies as a function of $v_{\rm max}$.
For the case of warm and mixed DM, we show that the discrepancy disappears,
even for luke-warm models that evade stringent bounds from the Lyman-$\alpha$
forest. Self-interacting DM scenarios can also provide a solution as long as
they lead to extended ($\gtrsim 1.5$ kpc) dark matter cores in the density
profiles of dwarf galaxies. Only models with velocity-dependent cross sections
can yield such cores without violating other observational constraints at
larger scales.
