  We present a general classification of simplified models that lead to dark
matter (DM) coannihilation processes of the form DM + X $\rightarrow$ SM$_1$ +
SM$_2$, where X is a coannihilation partner for the DM particle and SM$_1$,
SM$_2$ are Standard Model fields. Our classification also encompasses regular
DM pair annihilation scenarios if DM and X are identical. Each coannhilation
scenario motivates the introduction of a mediating particle M that can either
belong to the Standard Model or be a new field, whereby the resulting
interactions between the dark sector and the Standard Model are realized as
tree-level and dimension-four couplings. We construct a basis of coannihilation
models, classified by the $SU(3)_C\times SU(2)_L\times U(1)_Y$ quantum numbers
of DM, X and M. Our main assumptions are that dark matter is an electrically
neutral color singlet and that all new particles are either scalars, Dirac or
Majorana fermions, or vectors. We illustrate how new scenarios arising from
electroweak symmetry breaking effects can be connected to our electroweak
symmetric simplified models. We offer a comprehensive discussion of the
phenomenological features of our models, encompassing the physics of thermal
freeze-out, direct and indirect detection constraints, and in particular
searches at the Large Hadron Collider (LHC). Many novel signatures that are not
covered in current LHC searches are emphasized, and new and improved LHC
analyses tackling these signatures are proposed. We discuss how the
coannihilation simplified models can be used to connect results from all
classes of experiments in a straightforward and transparent way. This point is
illustrated with a detailed discussion of the phenomenology of a particular
simplified model featuring leptoquark-mediated dark matter coannihilation.
