  In this paper, we introduce the notion of Grothendieck enriched categories
for categories enriched over a sufficiently nice Grothendieck monoidal category
$\mathcal{V}$, generalizing the classical notion of Grothendieck categories.
Then we establish the Gabriel-Popescu type theorem for Grothendieck enriched
categories. We also prove that the property of being Grothendieck enriched
categories is preserved under the change of the base monoidal categories by a
monoidal right adjoint functor. In particular, if we take as $\mathcal{V}$ the
monoidal category of complexes of abelian groups, we obtain the notion of
Grothendieck dg categories. As an application of the main results, we see that
the dg category of complexes of quasi-coherent sheaves on a quasi-compact and
quasi-separated scheme is an example of Grothendieck dg categories.
