Iryna Egorova, Johanna Michor, and Gerald Teschl Inverse scattering transform for the Toda hierarchy with steplike finite-gap backgrounds (34K, LaTeX2e) ABSTRACT. We provide a rigorous treatment of the inverse scattering transform for the entire Toda hierarchy for solutions which are asymptotically close to (in general) different finite-gap solutions as $n\to\pm\infty$.