  To any dg-category $T$ (over some base ring $k$), we define a $D^{-}$-stack
$\mathcal{M}_{T}$ in the sense of \cite{hagII}, classifying certain
$T^{op}$-dg-modules. When $T$ is saturated, $\mathcal{M}_{T}$ classifies
compact objects in the triangulated category $[T]$ associated to $T$. The main
result of this work states that under certain finiteness conditions on $T$
(e.g. if it is saturated) the $D^{-}$-stack $\mathcal{M}_{T}$ is locally
geometric (i.e. union of open and geometric sub-stacks). As a consequence we
prove the algebraicity of the group of auto-equivalences of a saturated
dg-category. We also obtain the existence of reasonable moduli for perfect
complexes on a smooth and proper scheme, as well as complexes of
representations of a finite quiver.
