Fundamental properties of the dark matter can leave spectacular signatures on galactic and sub-galactic scales.  A flood of upcoming astrophysical data, coupled with rapid advancements in theory and simulation, open the opportunity to answer key questions about the nature of the dark sector.  In this talk, I will focus specifically on the question of whether new force carriers are allowed to mediate interactions between dark matter particles.  I will discuss how such self-interactions affect the properties of individual dwarf galaxies, as well as their population statistics around Milky Way-like hosts.  Using these results, I will show that current observational constraints strongly argue for self-interacting dark matter models that are velocity-dependent and lead to gravothermal collapse in the densest dwarf galaxies.  This remaining region of model space will be targeted with observational data on the abundance of dwarf galaxies, as well as cluster substructures.