This review examines how heat and spin flow in 1D quantum systems, especially those with special mathematical properties (integrable models) like the Heisenberg and Hubbard models. It covers recent theoretical and computational advances like Generalized Hydrodynamics, explaining how these systems can be surprisingly good conductors even when strongly interacting, sometimes displaying exotic phenomena like "superdiffusion."

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This is a comprehensive and well-written review that summarizes a significant amount of research on transport in 1D quantum lattice models. It clearly explains the theoretical background, key concepts, and various methods used to study transport properties, highlighting recent progress and open questions. The review primarily focuses on integrable systems, but also discusses non-integrable systems and their connection to real materials and experiments. While the review could benefit from a more detailed discussion of practical applications and experimental results, its strengths in clarity, comprehensiveness, and overall quality outweigh these limitations.