  Models in which the dark matter is very weakly coupled to the observable
sector may explain the observed dark matter density, either as a "superWIMP" or
as "asymmetric dark matter." Both types of models predict displaced vertices at
colliders, with a rich variety of possible phenomenology. We classify the cases
in which the decays can naturally occur inside particle detectors at the LHC,
with particular focus on the nontrivial scenarios where the decaying particle
is invisible. Identification of the position and timing of these invisible
displaced vertices significantly improves the prospects of reconstructing the
new physics in models such as supersymmetry. In many cases, reconstruction of
the visible products of the displaced decay can determine the dark matter mass,
allowing the dark matter density to be predicted from collider data.
