  We explore the possibility that bound states involving dark matter particles
could be detected by resonance searches at the LHC, and the generic
implications of such scenarios for indirect and direct detection. We
demonstrate that resonance searches are complementary to mono-jet searches and
can probe dark matter masses above 1 TeV with current LHC data. We argue that
this parameter regime, where the bound-state resonance channel is the most
sensitive probe of the dark sector, arises most naturally in the context of
non-trivial dark sectors with large couplings, nearly-degenerate
dark-matter-like states, and multiple force carriers. The presence of bound
states detectable by the LHC implies a minimal Sommerfeld enhancement that is
appreciable, and potentially also radiative bound state formation in the
Galactic halo, leading to large signals in indirect searches. We calculate
these complementary constraints, which favor either models where the
bound-state-forming dark matter constitutes a small fraction of the total
density, or models where the late-time annihilation is suppressed at low
velocities or late times. We present concrete examples of models that satisfy
all these constraints and where the LHC resonance search is the most sensitive
probe of the dark sector.
