One-dimensional quantum fluids are conventionally described using an effective hydrodynamic approach known as Luttinger liquid theory. As the principal simplification, a generic spectrum of the constituent particles is replaced by a linear one, which leads to a linear hydrodynamic theory. We show that to describe the measurable dynamic response functions one needs to take into account the nonlinearity of the generic spectrum, and thus of the resulting quantum hydrodynamic theory. This nonlinearity leads, for example, to a qualitative change in the behavior of the spectral function. The universal theory developed in this article is applicable to a wide class of fermionic, bosonic, and spin systems.