  We use ATLAS and CMS searches in the mono-jet + missing energy and
mono-photon + missing energy final state to set limits on the couplings of dark
matter to quarks and gluons. Working in an effective field theory framework we
compare several existing mono-jet analyses and find that searches with high p_T
cuts are more sensitive to dark matter. We constrain the suppression scale of
the effective dark matter-Standard Model interactions, and convert these limits
into bounds on the cross sections relevant to direct and indirect detection. We
find that, for certain types of operators, in particular spin-independent dark
matter-gluon couplings and spin-dependent dark matter-quark couplings, LHC
constraints from the mono-jet channel are competitive with, or superior to,
limits from direct searches up to dark matter masses of order 1 TeV. Comparing
to indirect searches, we exclude, at 90% C.L., dark matter annihilating to
quarks with the annihilation cross section of a thermal relic for masses below
~ 15-70 GeV, depending on the Lorentz structure of the effective couplings.
Mono-photon limits are somewhat weaker than mono-jet bounds, but still provide
an important cross check in the case of a discovery in mono-jets. We also
discuss the possibility that dark matter--Standard Model interactions at LHC
energies cannot be described by effective operators, in which case we find that
constraints can become either significantly stronger, or considerably weaker,
depending on the mass and width of the intermediate particle. We also discuss
the special case of dark matter coupling to the Higgs boson, and we show that
searches for invisible Higgs decays would provide superior sensitivity,
particularly for a light Higgs mass and light dark matter.
