  The Grothendieck construction is a classical correspondence between diagrams
of categories and coCartesian fibrations over the indexing category. In this
paper we consider the analogous correspondence in the setting of model
categories. As a main result, we establish an equivalence between suitable
diagrams of model categories indexed by $\mathcal{M}$ and a new notion of
\textbf{model fibrations} over $\mathcal{M}$. When $\mathcal{M}$ is a model
category, our construction endows the Grothendieck construction with a model
structure which gives a presentation of Lurie's $\infty$-categorical
Grothendieck construction and enjoys several good formal properties. We apply
our construction to various examples, yielding model structures on strict and
weak group actions and on modules over algebra objects in suitable monoidal
model categories.
