In part I, we will start by explaining what exactly derived manifolds are from various different perspectives, ranging from differential topology, supergeometry, and mathematical physics, and explain its role in BV-quantization. We will then provide a university property that uniquely characterizes such a theory, making direct links to DAG and graded supergeometry literature.

In part II, we will discuss how to extend this to a more general theory of derived differential geometry robust enough to deal with the infinite dimensional spaces arising in field theory, connections with infinite jet bundles and D-modules, and finally how to construct the derived moduli spaces of solutions to field equations of a (locally) Lagrangian field theory as a derived stack in this setting. Our main examples will be 3D Chern-Simons theory and 4D Yang-Mills.