A new approach to inverse scattering using the eigenfunctions of scattering operators is presented. This approach provides a unified framework that encompasses eigenfunction methods of focussing, diffraction tomography, and inverse scattering in arbitrary media. The eigenfunctions of scattering operators are used to specify source distributions that focus incident energy in the vicinity of inhomogeneities. The present inverse scattering approach represents unknown scattering media using products of numerically backpropagated fields of eigenfunctions. Recent progress in the mathematical theory of inverse scattering has suggested that these products are an appropriate basis for the reconstruction of unknown inhomogeneities. Because of the focussing property of scattering operators, these products also form an efficient basis for computational implementations of inverse scattering. The currently suggested approach has the further advantage of applicability to any medium for which a background Green's function can be determined. Computational results illustrate focussing of eigenfunctions on discrete and distributed scattering media, quantitative imaging analogous to diffraction tomography, and high-resolution inverse scattering in media with strongly scattering inhomogeneous backgrounds.