The motivation for this paper is the observation that most problems in physical oceanography and meteorology are concerned with mixtures of two or more constituents. Some diverse examples include seawater; turbidity currents, deforming seabeds containing pore fluids, and air containing all phases of water. In the last 15 years a revolution has occurred in the theoretical continuum approach to mixtures; however, none of this has found its way into the oceanographic or meteorological literature. Thus it is appropriate to compare models arising from this modern approach with the classical results. The major distinction between the two approaches is that the modern theory stipulates equations of motion for each constituent, whereas in the classical result it is assumed that the equations of motion for the mixture suffice for a description of the dynamics. The ramifications of this are explored in several diverse areas in oceanography and meteorology. It is shown that for the hydrothermal-dynamic description of seawater for purely dynamical purposes the distinction is not important. It is not clear that this is true for models of fine and microscale processes. The principal difficulty seems to be the parameterization of mixing by Fickian-type diffusion. Mixture theory provides an alternative that, for special situations, reduces to the Telegraph equation. For models of ocean seabed interactions it seems clear that the modern theory should be utilized. In this regard some general dynamical models for sediment transport and turbidity currents and viscous porous seabeds are developed. Also, the thermodynamics of a mixture of ice, water vapor, and condensate in the atmosphere is treated from the standpoint of the modern theory.