We investigate simplified models of new physics that can accommodate the measured value of
the anomalous magnetic moment of the muon and the relic density of dark matter. We define a
set of renormalizable, SU(2)$\times$U(1) invariant extensions of the Standard Model, each
comprising an inert $\mathbb{Z}_2$-odd scalar field and one or more vector-like pairs of
colorless fermions that communicate to the muons through Yukawa-type interactions. The new
sectors are classified according to their transformation properties under the Standard Model
gauge group and all models are systematically confronted with a variety of experimental
constraints: LEP mass bounds, direct LHC searches, electroweak precision observables, and
direct searches for dark matter. We show that scenarios featuring only one type of new
fermions become very predictive once the relic density and collider constraints are taken into
account, as in this case $g-2$ is not enhanced by chirality flip. Conversely, for models where
an additional source of chiral-symmetry violation is generated via fermion mixing, the
constraints are much looser and new precision experiments with highly suppressed systematic
uncertainties may be required to test the parameter space.