  We show that hydrodynamic diffusion is generically present in many-body
interacting integrable models. We 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 quasiparticles. We give exact expressions for the
diffusion coefficients. Our results apply to a large class of integrable
models, including quantum and classical, Galilean and relativistic field
theories, chains and gases in one dimension, such as the Lieb-Liniger model
describing cold atom gases and the Heisenberg quantum spin chain. We provide
numerical evaluations in the Heisenberg spin chain, both for the spin diffusion
constant, and for the diffusive effects during the melting of a small domain
wall of spins, finding excellent agreement with tDMRG numerical simulations.
