If dark matter couples to force mediators that are much lighter than itself, then its interactions manifest as long-ranged. This gives rise to non-perturbative effects, including the existence of bound states. The formation of stable or metastable dark matter bound states can affect the dark matter phenomenology very significantly, including the dark matter relic density and indirect detection signals, as well as the dark-matter self-scattering inside galaxies. I will give an overview of these effects in the context of dark matter production in the early universe, emphasising the important role of the Higgs, as well as the connection to unitarity.