Dark matter may have its own dark forces and interactions that are distinct from the Standard Model and unrelated the weak scale. To test this idea, galaxies and clusters of galaxies serve as cosmic colliders for measuring self-scattering among dark matter particles. Present constraints imply that if self-interactions are to solve the infamous core-cusp problem in dwarf galaxies, the scattering cross section must fall with energy/velocity to avoid cluster limits. To test this velocity dependence, I present new constraints on dark matter self-interactions at an intermediate scale with groups of galaxies. I also describe using mock observations from N-body simulations of self-interacting dark matter with baryons as a test of our methods. Lastly, I describe some recent work toward strongly-coupled theories of self-interacting dark matter, using tools borrowed from lattice QCD to compute its properties nonperturbatively.