  We show that simplified models used to describe the interactions of dark
matter with Standard Model particles do not in general respect gauge invariance
and that perturbative unitarity may be violated in large regions of the
parameter space. The modifications necessary to cure these inconsistencies may
imply a much richer phenomenology and lead to stringent constraints on the
model. We illustrate these observations by considering the simplified model of
a fermionic dark matter particle and a vector mediator. Imposing gauge
invariance then leads to strong constraints from dilepton resonance searches
and electroweak precision tests. Furthermore, the new states required to
restore perturbative unitarity can mix with Standard Model states and mediate
interactions between the dark and the visible sector, leading to new
experimental signatures such as invisible Higgs decays. The resulting
constraints are typically stronger than the 'classic' constraints on DM
simplified models such as monojet searches and make it difficult to avoid
thermal overproduction of dark matter.
