Graded manifolds have emerged as a unified geometric language for a broad spectrum of geometric and algebraic structures and as a powerful contemporary tool in mathematical physics. Differential graded manifolds, equipped with homological vector fields, turn out to be the correct language incorporating Lie algebroids, Courant algebroids and their higher structural analogues. At the same time they provide a systematic way to encode the structure of a wide class of quantum gauge field theories, especially through the BV/BRST formalism. This project aims to break new ground in graded geometry and make pioneering contributions in three interconnected research directions. First, we will investigate dg-manifolds with vanishing Atiyah class, aiming at three results of particular novelty: construction of the Atiyah class for Courant algebroids, of the Atiyah-Molino class for singular foliations and of the relative Atiyah class of a dg-morphism, the latter providing an obstruction theory for the existence of compatible connections along dg-maps. Another important objective in this direction is to generalize Kapranov structures in the setting of dg-bundles and to study the Atiyah class in graded symplectic geometry. Second, we will develop innovative tools in graded geometry with key results including (i) a robust notion of integration on graded manifolds, including Stokes' theorem and Poincaré duality, (ii) a Darboux theorem for graded symplectic manifolds of arbitrary degree and an analogue of Weinstein’s splitting theorem, and (iii) to obtain the category of complex graded manifolds via a suitable generalization of the Newlander–Nirenberg theorem and interesting examples thereof. Finally, we will harness this mathematical toolkit to give feedback to physics, exploring applications of graded geometry in gravity, gauge theory, sigma-models and quantization.