DG manifolds are a useful geometric notion that unifies many important structures such as homotopy Lie algebras, foliations, and complex manifolds. In particular, the Todd class of DG manifolds extends both the Todd class of a complex manifold and the Duflo element in Lie theory. DG manifolds also play an important role in the study of the differential geometry of singular spaces. In this series of lectures, we will introduce DG manifolds and discuss various applications in geometry and higher structures. We will describe noncommutative calculi for DG manifolds. In particular, we will prove a Duflo-Kontsevich type theorem for DG manifolds. The classical Duflo theorem in Lie theory and the Kontsevich theorem regarding the Hochschild cohomology of complex manifolds can both be derived as special cases of this Duflo-Kontsevich type theorem for DG manifolds. We will also discuss its application in derived differential geometry, which deals with singularities arising from zero loci or intersections of submanifolds. This is mainly based on joint work with Kai Behrend, Ruggero Bandiera, Zhuo Chen, Hsuan-Yi Liao, Rajan Mehta, Seokbong Seol, Mathieu Stienon, and Maosong Xiang.