This paper proves some results about three types of DG categories over algebraic stacks \(\mathcal{Y}\): QCoh(\(\mathcal{Y}\)) the DG category of quasi-coherent sheaves on \(\mathcal{Y}\), IndCoh(\(\mathcal{Y}\)) the ind-completion of the full subcategory Coh(\(\mathcal{Y}\)) of QCoh(\(\mathcal{Y}\)) consisting of bounded complexes with coherent cohomology sheaves, and D-mod(\(\mathcal{Y}\)) the DG category of D-modules on \(\mathcal{Y}\). Those DG categories are not independent; results about IndCoh\((\mathcal{Y})\) will follow from those on QCoh\((\mathcal{Y})\), and results about D-mod(\(\mathcal{Y})\) will follow from using some adjoint pair of functors that we have by virtue of the existence of a left adjoint to some conservative forgetful functor D-mod(\(\mathcal{Y}) \rightarrow \text{IndCoh}(\mathcal{Y})\). The DG categories considered in this paper are cocomplete, in which case we can study whether they have compact objects and hence whether they are compactly generated. The algebraic stacks considered are of finite type over a field \(k\) of characteristic zero and are assumed to be such that the automorphism groups of their geometric points are affine. Such stacks are referred to as QCA algebraic stacks. For a prestack \(\mathcal{Y}\) and a morphism \(p_{\mathcal{Y}}: \mathcal{Y} \rightarrow \) pt, \(\Gamma(\mathcal{Y}, -):=(p_{\mathcal{Y}})_*\). By stratifying QCA algebraic stacks \(\mathcal{Y}\) by locally closed substacks, the authors prove that there is an integer \(n\) depending only on \(\mathcal{Y}\) such that \(H^i(\Gamma(\mathcal{Y}, \mathcal{F}))=0\) for all \(i>n\) and all \(\mathcal{F} \in \text{QCoh}(\mathcal{Y})^{\leq 0}\), from which they infer that the functor QCoh(\(\mathcal{Y}) \rightarrow \text{Vect}: \mathcal{F} \mapsto \Gamma(\mathcal{Y}, \mathcal{F})\) is continuous. From that result they further consider those QCA algebraic stacks that in addition have the property of being what they call locally almost of finite type, meaning that they have an atlas with a DG scheme that can be covered by affines Spec(\(A\)) such that \(H^0(A)\) is a finitely generated algebra over \(k\) and each \(H^{-i}(A)\) is a finitely generated \(H^0(A)\)-module. From the above result they prove that for a QCA algebraic stack locally almost of finite type \(\mathcal{Y}\) then IndCoh\((\mathcal{Y})\) and D-mod(\(\mathcal{Y}\)) are compactly generated. There is no such result for QCoh(\(\mathcal{Y}\)) but the authors first prove that the DG category IndCoh\((\mathcal{Y})\) is dualizable, from which they also prove that if in addition the algebraic stack \(\mathcal{Y}\) is eventually coconnective, QCoh(\(\mathcal{Y})\) is dualizable as well. The authors also prove that for \(\mathcal{Y}\) a quasi-compact stack, the functor \(\Gamma_{\text{dR}}(\mathcal{Y}, -)\) is continuous if and only if \(\mathcal{Y}\) is safe, meaning that the neutral connected component of all of its geometric points’ automorphism group is unipotent. The authors also fix the problem of having the functor \(\pi_{\text{dR},*}\) of direct image on D-modules not being continuous by defining what they call a renormalized direct image functor \(\pi_{\blacktriangle}\), the dual of \(\pi^!\) where \(\pi: \mathcal{Y}_1 \rightarrow \mathcal{Y}_2\) is a map of QCA algebraic stacks. Such a definition is made possible thanks to a generalization of Verdier duality for D-modules on QCA algebraic stacks locally almost of finite type. Such a renormalized direct image functor is continuous, and they prove that on safe objects we have an isomorphism \(\pi_{\blacktriangle} \rightarrow \pi_{\text{dR},*}\). In addition to these results other peripheral but still very interesting results are presented. For lack of sufficient references in the literature regarding the concepts covered in the paper, the authors liberally provide some highly valuable background material. The exposition does not lack in interesting examples and counter-examples where needed.