A theory for the flow and non-linear diffusion effects in mixtures of fluids is formulated based upon hydrodynamical considerations. It is assumed that each point of the mixture is occupied simultaneously by all constituents in given portions. The motion of each constituent is governed by the usual equations of motion and continuity. The mechanical properties of each component are specified by means of constitutive equations for the stresses; diffusion effects are accounted for by means of a body force acting on each constituent which depends upon the composition and relative motion of the substances in the mixture. The theory is extended to deal with the diffusion of a mixture of fluids through a rigid solid. The theory is applied to a number of steady-state problems involving non-Newtonian fluids including the diffusion of a fluid through a rigid plate, the laminar flow of a mixture and the flow of a mixture between rotating cylinders. The propagation of plane waves through a homogeneous mixture of viscous fluids at rest is also examined.