Gravitational probes of dark matter may provide the first hints of beyond-the-standard-model physics in the dark sector. For example, if a component of the dark matter has dissipative interactions, it could collapse to form a thin dark disk in our Galaxy coincident with the baryonic disk. It has been suggested that dark disks could explain a variety of observed phenomena. Using the first data release from the Gaia space observatory, I will present the results of a search for a dark disk via its effect on stellar kinematics in the Milky Way. I will discuss how our new limits disfavor the presence of a thin dark disk. I will then present a new method for using stellar kinematic data to search for low-mass substructure in the Milky Way’s dark matter halo, at masses ~10^7 solar masses or below. Applications of this method to future datasets from observatories such as Gaia and LSST could shed light on a variety of dark matter models, including warm dark matter, fuzzy dark matter, and self-interacting dark matter.