Conservation laws, heirarchies, scattering theory and Backlund transformations are known to be the building blocks of integrable partial differential equations. We identify these as facets of a theory of Poisson group actions, and apply the theory to the ZS-AKNS nxn heirarchy (which includes the non-linear Schrodinger equation, modified KdV, and the n-wave equation). We first find a simple model Poisson group action that contains flows for systems with a Lax pair whose terms all decay on R. Backlund transformations and flows arise from subgroups of this single Poisson group. For the ZS-AKNS nxn heirarchy defined by a constant a ∈ u(n), the simple model is no longer correct. The a determines two separate Poisson structures. The flows come from the Poisson action of the centralizer Ha of a in the dual Poisson group (due to the behavior of e at infinity). When a has distinct eigenvalues, Ha is abelian and it acts symplectically. The phase space of these flows is the space Sa of left cosets of the centralizer of a in D−, where D− is a certain loop group. The group D− contains both a Poisson subgroup corresponding to the continuous scattering data, and a rational loop group corresponding to the discrete scattering data. The Ha-action is the right dressing action on Sa. Backlund transformations arise from the action of the simple rational loops on Sa by right multiplication. Various geometric equations arise from appropriate choice of a and restrictions of the phase space and flows. In particuar, we discuss applications to the sine-Gordon equation, harmonic maps, Schrodinger flows on symmetric spaces, Darboux orthogonal coordinates, and isometric immersions of one space-form in another. 1 Research supported in part by NSF Grant DMS 9626130 2 Research supported in part by Sid Richardson Regents’ Chair Funds, University of Texas system