In this series of three lectures I will discuss non-linear quantum hydrodynamics in quantum electronic liquids. I will concentrate on three of the most important systems:

• Degenerate Fermi gas in one spatial dimension,
• Edge state in the Fractional Quantum Hall Effect,
• Luttinger Liquid.

I will discuss

• why Fermi statistics causes non-equilibrium quantum liquids to be essentially non-linear,
• instabilities and singularities arising as results of the non-linear nature of quantum liquids,
• mathematical methods for the analysis of singularities in the dynamics of quantum liquids,
• relations to random matrix theory and to perturbed conformal field theory.