We perform a general study of the relic density and LHC constraints on simplified models where the dark matter coannihilates with a strongly interacting particle X. In these models, the dark matter depletion is driven by the self-annihilation of X to pairs of quarks and gluons through the strong interaction. The phenomenology of these scenarios therefore only depends on the dark matter mass and the mass splitting between dark matter and X as well as the quantum numbers of X. In this paper, we consider simplified models where X can be either a scalar, a fermion or a vector, as well as a color triplet, sextet or octet. We compute the dark matter relic density constraints taking into account Sommerfeld corrections and bound state formation. Furthermore, we examine the restrictions from thermal equilibrium, the lifetime of X and the current and future LHC bounds on X pair production. All constraints are comprehensively presented in the mass splitting versus dark matter mass plane. While the relic density constraints can lead to upper bounds on the dark matter mass ranging from 2 TeV to more than 10 TeV across our models, the prospective LHC bounds range from 800 to 1500 GeV. ... mehrA full coverage of the strongly coannihilating dark matter parameter space would therefore require hadron colliders with significantly higher center-of-mass energies.