Hydrodynamics is a powerful theory for describing the emergent out-of-equilibrium dynamics at large scales in space and time. The Navier-Stokes equation is an example of a hydrodynamic equation, with complex and beautiful phenomenology. But it is just an example, and the basic precepts of hydrodynamics can be applied to a much wider set of many-body systems, quantum and classical. An interesting family is that of integrable models: they admit a large number of conservation laws, and this strongly affects their thermalisation and out-of-equilibrium properties. Integrability is in a sense the most fundamental attribute of quasi-one-dimensional physics, and remains relevant up to measurable times in many experimentally realised, non-integrable models. The theory adapting hydrodynamics to integrability, dubbed "generalised hydrodynamics" (GHD), has been developed in recent years. It turns out to be extremely general, applicable to quantum and classical chains, field theories and gases.

These lectures will cover the fundamental concepts of this theory. I will start with an overview of what, fundamentally, hydrodynamics is about. I will then describe the most important aspects of integrability, from which I will derive the basic GHD equations. If time permits, I will explain one or two applications or more advanced concepts, such as the Riemann problem, correlation functions, large deviations, and diffusion. In order to have clear examples, I will focus on two simple models: the quantum Lieb-Liniger model, and the classical hard rod gas. The GHD of the Lieb-Liniger model has even been verified experimentally!

Only very basic knowledge of hydrodynamics and integrability is assumed, and I will keep everything as non-technical as possible.