One might expect high-temperature transport to be incoherent and diffusive. However, many experimentally relevant one-dimensional systems, such as the Heisenberg and Hubbard models, feature long-lived quasiparticles that give rise to anomalous transport. I will present a quantitative theory of high-temperature transport in such systems, in terms of a dense gas of stable quasiparticles that scatter elastically off one another. This theory resolves some puzzles, but many remain; I will discuss how observables beyond linear response, which are straightforward to measure in ultracold atomic gases, can shed light on these.