  Studies of dark matter models lie at the interface of astrophysics,
cosmology, nuclear physics and collider physics. Constraining such models
entails the capability to compare their predictions to a wide range of
observations. In this review, we present the impact of global constraints to a
specific class of models, called dark matter simplified models. These models
have been adopted in the context of collider studies to classify the possible
signatures due to dark matter production, with a reduced number of free
parameters. We classify the models that have been analysed so far and for each
of them we review in detail the complementarity of relic density, direct and
indirect searches with respect to the LHC searches. We also discuss the
capabilities of each type of search to identify regions where individual
approaches to dark matter detection are the most relevant to constrain the
model parameter space. Finally we provide a critical overview on the validity
of the dark matter simplified models and discuss the caveats for the
interpretation of the experimental results extracted for these models.
