Stability of interfaces moving as a result of phase transformation or mass transfer is considered. Effects of transport and of the density difference between phases known to significantly influence stability during solidification and combustion respectively are here studied theoretically in fluid systems. The analysis indicates that instability is possible during vaporization of liquids under vacuum and perhaps during growth of cavitation bubbles. But except when interfacial tension is very low, instability is unlikely during mass transfer in liquid-liquid systems because the density difference between phases is small. A density difference can also produce cellular convection at an unstable interface during solidification.