The Goodwillie derivatives of a functor from based spaces to spectra possess additional structure that allows the Taylor tower of the functor to be reconstructed. I will describe this structure as a 'module' over the 'pro-operad' formed by the Koszul duals of the little disc operads. For certain functors this structure arises from an actual module over the little L-discs operad for some L. In particular, this is the case for functors that are left Kan extensions from a category of 'pointed framed L-dimensional manifolds' (which are examples of the zero-pointed manifolds of Ayala and Francis). As an application I will describe where Waldhausen's algebraic K-theory of spaces fits into this picture. This is joint work with Greg Arone (and, additionally, with Andrew Blumberg for the application to K-theory).