We describe a new mathematical framework for solving a wide variety of rendering problems based on a non-linear integral scattering equation. This framework treats the scattering functions of complex aggregate objects as first-class rendering primitives; these scattering functions accurately account for all scattering events inside them. We also describe new techniques for computing scattering functions from the composition of scattering objects. We demonstrate that solution techniques based on this new approach can be more efficient than previous techniques based on radiance transport and the equation of transfer and we apply these techniques to a number of problems in rendering scattering from complex surfaces.