In this talk I will present some results concerning the analysis of the existence of finite energy weak solutions of the Navier-Stokes-Korteweg equations, which are a very general class of compressible fluids models for various physical situations, such as the dynamic of a viscous compressible fluid with degenerate viscosity and capillarity tensor or quantum fluids. These kind of models are also useful to study the dynamic of fluids near vacuum regions. A general theory of global existence is still missing, however for some particular cases of physical interest, it is possible to prove global existence of weak solutions. In particular, I will present two results regarding the case when the capillarity coefficient is constant and when the capillarity coefficient arises from the Bohm potential. The talk is based on a series of joint works with Paolo Antonelli (GSSI - Gran Sasso Science Institute, L’Aquila)