The nature of dark matter remains one of the great puzzles of fundamental physics, with potential connections to a plethora of deep questions. While dark matter is often assumed to be collisionless, dark matter particles may interact with each other through either the known forces of the Standard Model of particle physics, or new "dark" forces, while remaining consistent with current constraints. In either case, dark matter may form bound states similar to those we observe in ordinary matter. I will discuss a range of possible implications of the existence of such bound states, including observational signatures in gamma-ray telescopes, and modifications to the history of the cosmos allowing for strikingly heavy dark matter in strongly-interacting dark sectors.