These lecture notes introduce a modern, controlled way to describe fluctuating hydrodynamics — the random, slow motion that appears in essentially any local many‑body system at nonzero temperature. The author argues that an effective field theory (EFT) approach captures the universal, late‑time behavior of correlation functions and transport, even when the microscopic dynamics are complicated or strongly interacting.

The lectures collect the organizing principles behind equilibrium and out‑of‑equilibrium dynamics. A central idea is a modern construction of hydrodynamic EFTs framed in terms of “strong‑to‑weak” spontaneous symmetry breaking. The notes draw examples across physics, from spin chains to relativistic quantum field theories, and treat cases with generalized symmetries and ’t Hooft anomalies. They also discuss constraints on transport coefficients — viewed as Wilson coefficients, the few free parameters of an EFT — both in the continuum and on the lattice. The material is based on talks given at TASI and the Boulder Summer School in 2025.

At a high level the EFT idea is simple. If a system thermalizes locally, only a small set of slow quantities matter at late times: conserved densities like energy or charge. An effective action or generating functional then organizes observables in a derivative expansion. The leading terms fix equilibrium quantities such as pressure and susceptibilities. The unknown coefficients in the expansion (Wilson coefficients) encode transport properties like diffusion or conductivity. Because the EFT focuses on long times and long distances, it can be predictive even when the microscopic model is hard to solve.

This perspective matters because hydrodynamics is extremely universal. It applies to metals, magnets, spin chains, quantum field theories and many other systems. The EFT framework gives a systematic way to compute real‑time correlators and nonlinear responses to weak probes. That is especially useful when standard perturbation theory breaks down at late times or when interactions are strong.