Let M be a smooth supermanifold and n a supertranslation algebra. Manin defined a superconformal structure of type n on M as an odd maximal dimensional distribution on M whose symbol is isomorphic to n at every point. The goal of this talk will be to illustrate an approach to supersymmetric field theories wherein spacetime is replaced with a dg-ringed space determined by such a datum. We'll first show how this perspective gives a natural geometric interpretation to the so-called pure spinor superfield formalism - a set of techniques for universal constructions of off-shell supermultiplets. Next, we will turn to studying the tangent sheaf of these dg ringed spaces. These are sheaves of dg Lie algebras interesting in their own right; in physical examples H^0 returns the global superconformal algebras classified by Nahm and Kac-van de Leur whilst H^1 reproduces conformal supergravity multiplets. This talk is based on joint work with Fabian Hahner, Ingmar Saberi, and Brian Williams.