These lecture notes provide a concise review of the mathematical theory for Quantum Hydrodynamics (QHD) equations, describing the evolution of a compressible, inviscid, barotropic flow subject to a stress tensor depending on the mass density and its derivatives. Such fluid dynamical systems describe several phenomena where quantum effects remain relevant also on a macroscopic scale. Most of these models are deeply interconnected with nonlinear Schrödinger (NLS) type equations through the so-called Madelung transformation. Our approach exploits this analogy to construct and analyze global-in-time, finite energy weak solutions to the QHD system. It is known that the Madelung transformation becomes singular close to vacuum, where the mass density vanishes. We introduce an alternative approach based on the polar decomposition of a wave function that rigorously establishes the relation between NLS evolutions and QHD systems. After briefly reviewing the literature concerning NLS equations, we introduce the main relevant objects for quantum fluid mechanics. We then discuss notions of weak solutions defined through physically relevant entropy functionals. We first analyze the polar factorization method and its inverse given by the wave function lifting. By exploiting these tools, we provide some existence results for solutions to various QHD systems, including the collisional model that does not have an analog for NLS-type evolutions. We then study the stability properties of general weak solutions. We derive dispersive-type estimates that provide information on the asymptotic behavior for large times. Moreover, by introducing a novel entropy functional that controls a generalized chemical potential, we identify a class of weak solutions for which it is possible to prove a compactness result for the one-dimensional QHD system with large data.