The field of hydrodynamics of quantum systems has experienced a revival in the past decade, as an effective theory describing how many-body quantum systems evolve from local to global equilibrium. This has been largely driven by the advent of new experimental platforms, from strongly interacting ultracold gases to pristine solid-state systems with strong interactions and long mean free times. Hydrodynamics is particularly rich for low-dimensional fluids, featuring transport anomalies such as long-time tails, and proximity of many realistic systems to integrability. In this talk, I will focus on the hydrodynamics of systems close to integrable limits, featuring infinitely-many approximate conservation laws and long-lived quasiparticle excitations. I will review recent successes of this theory, and use it to argue that isotropic magnets in one dimension can exhibit a breakdown of Fick’s diffusion law, corresponding to anomalous, “superdiffusive” transport properties. I will discuss recent experiments probing these transport anomalies and conclude with open questions that they raise.