  We study renormalisable models with minimal field content that can provide a
viable Dark Matter candidate through the standard freeze-out paradigm and,
simultaneously, accommodate the observed anomalies in semileptonic $B$-meson
decays at one loop. Following the hypothesis of minimality, this outcome can be
achieved by extending the particle spectrum of the Standard Model either with
one vector-like fermion and two scalars or two vector-like fermions and one
scalar. The Dark Matter annihilations are mediated by $t$-channel exchange of
other new particles contributing to the $B$-anomalies, thus resulting in a
correlation between flavour observables and Dark Matter abundance. Again based
on minimality, we assume the new states to couple only with left-handed muons
and second and third generation quarks. Besides an ad hoc symmetry needed to
stabilise the Dark Matter, the interactions of the new states are dictated only
by gauge invariance. We present here for the first time a systematic
classification of the possible models of this kind, according to the quantum
numbers of the new fields under the Standard Model gauge group. Within this
general setup we identify a group of representative models that we
systematically study, applying the most updated constraints from flavour
observables, dedicated Dark Matter experiments, and LHC searches of leptons
and/or jets and missing energy, and of disappearing charged tracks.
