Explore superdiffusion, subdiffusion, and integrability in non-equilibrium many-body physics through this conference talk recorded at the 2021 Non-Equilibrium Universality in Many-Body Physics KITP Conference. Delve into the quasiparticle picture of integrable systems, focusing on the XXZ spin chain as a case study. Examine integrability-breaking and its implications for ballistic transport. Gain insights into diffusion through intuitive arguments and alternative framings. Discover how quantum simulators have revolutionized the field, bridging gaps between statistical physics, AMO, condensed matter, and high-energy physics. Investigate universal aspects of many-body systems far from equilibrium, exploring novel phases of matter and their associated universality classes beyond traditional equilibrium paradigms.