Abstract: Given two smooth maps of manifolds and if they are not transverse, the fibered product may not exist, or may not have the correct cohomological properties. Thus lack of transverality obstructs many natural constructions in topology and differential geometry. Derived manifolds generalize the concept of smooth manifolds to allow arbitrary (iterative) intersections to exist as smooth objects, regardless of transversality. In this talk we will provide some motivation for the development of the theory of derived manifolds, ultimately formulating a universal property. We will then explain how this universal property yields several concrete models. We will also discuss work in progress with D. Roytenberg whose goal is to give an accessible model based of the geometry of differential graded manifolds. Time permitting, we will discuss some aspects of our joint work with O. Gwilliam on applications of this framework to classical field theory.