The modern foundations of Derived Algebraic Geometry are well established by the seminal works of Lurie and Toën-Vezzosi. Following their philosophy, various (related) approaches to a differential version of derived geometry have also been put forth by Spivak, Borisov-Noël, Carchedi-Roytenberg, and Joyce, among others. For derived differential geometry in its most general incarnation however, several key constructions and results available in the algebraic and analytic contexts have not yet been verified. The goal of this talk will be to explain how to fill in some of these gaps: I will introduce an $\infty$-category of derived smooth manifolds that enjoys a `universal property of derived geometry', and we will see how the yoga of derived geometric stacks carries over neatly to the differentiable context. If time permits, we will touch upon deformation theory of derived stacks and state Lurie's version of Artin's representability theorem: a powerful criterion for verifying that, for instance, moduli spaces of solutions of various nonlinear elliptic PDE's are presented by stacky quotients of derived manifolds. Our hope is that these results will facilitate a systematic study of the derived symplectic geometry of (possibly very singular) moduli spaces of Euler-Lagrange equations in Quantum Field Theories, which have their natural home in derived smooth geometry.