We utilize a modern continuum mechanics framework to reconsider an old problem for fluid interfaces, also addressed by Maxwell and van der Waals. We prove that their results need not be valid necessarily. This conclusion is arrived at as a consequence of questioning the existence of thermodynamic potentials and the validity of usual thermodynamic ,relations within unstable (spinodal) regions. One central result is that Maxwell's equal area rule needs not be valid and certain statistical models are shown to be internally inconsistent. Precise conditions for the validity of Maxwell's rule and the variational theory of van der Waals are established in terms of the coefficients defining the interfacial stress. Finally, a generalized continuum thermodynamics framework is developed which provides an alternative derivation of van der Waals variational theory and properly extends it to dynamic situations. However, other possibilities exist which allow a thermodynamics of fluid interfaces not necessarily restricted by the conditions of Maxwell and van der Waals. The results of the paper could be viewed as a convincing argument to utilize this framework (with the necessary modifications) to interpret more complex phenomena of phase transformations.