  This review summarizes recent advances in our understanding of anomalous
transport in spin chains, viewed through the lens of integrability. Numerical
advances, based on tensor-network methods, have shown that transport in many
canonical integrable spin chains -- most famously the Heisenberg model -- is
anomalous. Concurrently, the framework of generalized hydrodynamics has been
extended to explain some of the mechanisms underlying anomalous transport. We
present what is currently understood about these mechanisms, and discuss how
they resemble (and differ from) the mechanisms for anomalous transport in other
contexts. We also briefly review potential transport anomalies in systems where
integrability is an emergent or approximate property. We survey instances of
anomalous transport and dynamics that remain to be understood.
