We use Lurie's theory of derived geometry to formulate a category of (quasi-smooth) Derived Manifolds. This category contains the category of smooth manifolds, but is also closed under non-transverse intersections. More generally, the solution to any smooth equation on a manifold is a derived manifold, even though its underlying set may be very singular.
Despite this generality, we can prove that every derived manifold has a fundamental class. Thus we may do intersection theory on derived manifolds. We will sketch this proof, as well as an imbedding theorem and a Thom-Pontrjagin theorem for derived manifolds.