  The last decade has witnessed an impressive progress in the theoretical
understanding of transport properties of clean, one-dimensional quantum lattice
systems. Many physically relevant models in one dimension are Bethe-ansatz
integrable, including the anisotropic spin-1/2 Heisenberg (also called spin-1/2
XXZ chain) and the Fermi-Hubbard model. Nevertheless, practical computations
of, for instance, correlation functions and transport coefficients pose hard
problems from both the conceptual and technical point of view. Only due to
recent progress in the theory of integrable systems on the one hand and due to
the development of numerical methods on the other hand has it become possible
to compute their finite temperature and nonequilibrium transport properties
quantitatively. Most importantly, due to the discovery of a novel class of
quasilocal conserved quantities, there is now a qualitative understanding of
the origin of ballistic finite-temperature transport, and even diffusive or
super-diffusive subleading corrections, in integrable lattice models. We shall
review the current understanding of transport in one-dimensional lattice
models, in particular, in the paradigmatic example of the spin-1/2 XXZ and
Fermi-Hubbard models, and we elaborate on state-of-the-art theoretical methods,
including both analytical and computational approaches. Among other novel
techniques, we discuss matrix-product-states based simulation methods,
dynamical typicality, and, in particular, generalized hydrodynamics. We will
discuss the close and fruitful connection between theoretical models and recent
experiments, with examples from both the realm of quantum magnets and ultracold
quantum gases in optical lattices.
