Abstract: Many physical systems with chaotic microscopic dynamics display remarkably regular macroscopic behavior. For example, gases made of many interacting particles are well described, at large scales, by familiar hydrodynamic equations such as those of Euler or Navier–Stokes. These systems mix efficiently and have only a few conserved quantities, like energy and momentum.Integrable systems, on the other hand, behave very differently. They possess infinitely many conserved quantities, which prevent the usual mixing and lead to distinct large-scale evolution laws—often referred to as generalized hydrodynamics. Examples include one-dimensional hard-rod gases, the Toda lattice, the Korteweg–de Vries (KdV) and nonlinear Schrödinger equations, and even certain cellular automata such as the box–ball system.While these ideas have been extensively explored in the physics literature, rigorous mathematical understanding is still in its early stages. In this talk, I will discuss what is known, what is conjectured, and some of the recent progress toward a mathematical theory of macroscopic behavior in integrable dynamics.