The authors derive the Navier-Stokes-Allen-Cahn (NSAC), Navier-Stokes-Cahn-Hilliard (NSCH) and Navier-Stokes-Korteweg (NSK) systems of equations starting from general two-phase fluid flows and they establish links between these systems. They indeed consider fluid flows described through the variations of \((\rho ,\theta ,\chi ,u)\) on \([0,T]\times \overline{\Omega }\) where \(\rho \) is the density of the fluid, \(\theta \) its temperature and \(u\) its mass-averaged velocity, and \(\chi \) denotes the concentration of one phase. Here \(\Omega \) is a domain of \(\mathbb{R}^{n}\). The authors introduce the Helmholtz and Gibbs energies for each phase and mixing Helmholtz and Gibbs energies taking the same value. They especially consider four classes of fluids for which either Raoult or Dalton law are valid and satisfying properties of the Gibbs energies of the two phases. The authors start with the Navier-Stokes-Allen-Cahn equations for which they write the classical conservations of mass, momentum and energy and for which they add an equation \(\partial _{t}(\rho \chi )+\nabla \cdot (\rho \chi u)-J=0\) which expresses the transformation between the two phases, \(J\) being a transformation rate. They then derive properties of this system and they derive the expressions of Ericksen’s stress tensor \(\mathbf{C}_{E}=-\nabla \chi \otimes \frac{\partial }{\partial \nabla \chi }(\rho G)\) and of the transformation rate \(J_{AC}=\frac{\theta }{\varepsilon }(-\rho \frac{ \partial }{\partial \chi }(\frac{1}{\theta }G)+\nabla \cdot (\rho \frac{ \partial }{\partial \nabla \chi }(\frac{1}{\theta }G))\) in terms of Gibbs energy. Changing the structure of the transformation rate, the authors then derive the Navier-Stokes-Cahn-Hilliard equations. Here they link their results to that obtained by J. Lowengrub and L. Truskinovsky in [Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 454, No. 1978, 2617–2654 (1998; Zbl 0927.76007)]. In this section, they finally derive the Navier-Stokes-Korteweg equations now considering an internal energy whose expression is related to the Helmholtz energy. Considering the case of two incompressible phases of different temperature-independent specific volumes, the authors prove that the NSAC or NSCH equations may be written as a special NSK system for appropriate choices of the Helmhotz energy and of the viscous stress. The paper ends with the derivation of energy balances in the isothermal case for the NSAC, NSCH and NSK systems.