It will be reviewed briefly various physical models leading to a description based on Quantum Hydrodynamics: Superfluidity, BEC, superconductivity, semiconductors and there will be recalled various derivations of the PDE system. The main result (joint with P. Antonelli) shows the global existence of "irrotational" weak solutions with the sole assumption of finite energy, without any smallness or any further smoothness of the initial data. The approach is based on various tools, namely the wave functions polar decomposition, the construction of approximate solution via a fractional steps method which iterates a Schrödinger Madelung picture with a suitable wave function updating mechanism. Therefore several a priori bounds of energy, dispersive and local smoothing type, allow to prove the compactness of the approximating sequences. A different approach can be used to study small disturbances of subsonic (in the QHD sense) steady states. Some improvements may are shown to be possible in the 2-D analysis.