  Understanding the general principles underlying strongly interacting quantum
states out of equilibrium is one of the most important tasks of current
theoretical physics. With experiments accessing the intricate dynamics of
many-body quantum systems, it is paramount to develop powerful methods that
encode the emergent physics. Up to now, the strong dichotomy observed between
integrable and non-integrable evolutions made an overarching theory difficult
to build, especially for transport phenomena where space-time profiles are
drastically different. We present a novel framework for studying transport in
integrable systems: hydrodynamics with infinitely-many conservation laws. This
bridges the conceptual gap between integrable and non-integrable quantum
dynamics, and gives powerful tools for accurate studies of space-time profiles.
We apply it to the description of energy transport between heat baths, and
provide a full description of the current-carrying non-equilibrium steady state
and the transition regions in a family of models including the Lieb-Liniger
model of interacting Bose gases, realized in experiments.
