This dissertation explores the application of the effective field theory (EFT) formalism to the study of finite density and hydrodynamical systems. Such systems have a preferred reference frame and thus spontaneously break boost invariance, and possibly more symmetries. Adopting a relativistic notation, one can describe their low-energy, long-distance behavior in terms of the associated Goldstone bosons.

In the first part, we study energy flux correlators and other event shapes on states created by the insertion of operators with large global U(1) charge, in a large class of conformal field theories. For this class of theories, the large charge sector can be described by the EFT of a superfluid, that emerges naturally when performing a Weyl transformation to map the problem to the cylinder. We develop a general framework to study these event shapes in the EFT at leading and next-to-leading order.

In the second part, we study the effective field theory of perfect fluids. Due to the mathematical obstruction to the formulation of this EFT in terms of the standard hydrodynamical degrees of freedom, such as density, pressure, or fluid velocity, there exist different formulations reproducing Euler's equation. We perform an in-depth study of the different formulations, showing in particular their subtle inequivalence. We then study the problem of quantizing these theories. The difficulty comes from the presence of turbulences at the classical level, which translates into UV-IR mixing, as well as an infinitely degenerate spectrum, at the quantum level.

In the third and last part, we study finite temperature dissipative systems. To expand the formalism of effective field theory of finite density systems to include dissipative effects, one need to resort to the Schwinger-Keldysh formalism. Focusing on the case of non-abelian charge transport, we study two approaches put forward in the literature using the coset construction, proving their equivalence.