I will show how hydrodynamic diffusion is generically present in many-body one-dimensional interacting quantum and classical integrable models. I will extend the recently developed generalised hydrodynamic (GHD) to include terms of Navier-Stokes type which lead to positive entropy production and diffusive relaxation mechanisms. These terms provide the subleading diffusive corrections to Euler-scale GHD for the large-scale non-equilibrium dynamics of integrable systems, and arise due to two-body scatterings among the quasiparticles of the model. Moreover I will show how with some particular choice of Hamiltonian interactions, the diffusion constant relative to the spin or charge degrees of freedom diverges, signaling the presence of super diffusive transport.