The abundance and internal structure of dark matter halos and subhalos on sub-galactic scales, below 10^9 solar masses, depend on the formation mechanism, mass, and possible self-interactions of the dark matter particle(s). As such, inferences of the properties of low-mass halos can be recast as constraints on fundamental dark matter physics. Strong gravitational lensing by galaxies provides a direct, purely gravitational means of inferring the properties of dark matter structure down to roughly 10^7 solar masses with existing data. University of Toronto's Daniel Gilman will describe how we can analyze samples of quadruple-image strong lenses to infer properties of dark substructure, and constrain several classes of dark matter theories, including warm, ultra-light, and self-interacting dark matter, that make unique predictions for the properties of the unseen dark structure of the universe.