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Accurate models for multicomponent transport are a prerequisite for the design of many industrial processes and the interpretation of experiments. Present theories, stemming from the statistical-mechanics developments of Chapman-Enskog, Zhdanov-Kagan-Sazykin, and Bearman-Kirkwood are shown to apply only to systems with low shear. These theories are not able to describe gaseous counterdiffusion in capillaries, such as in the experiments of Remick and Geankoplis, and the salt diffusion experiment in a simple cylinder by Fick. A simple experiment shows that the irreversible thermodynamics approach by De Groot-Mazur and Hirschfelder-Curtiss-Bird provides inconsistencies. The cause for this is the common development of the theories as perturbations superposed on the mass-averaged velocity. A new solution to the Boltzmann equation is developed for monatomic dilute gases, based on nonequilibrium trial functions, in which the velocity distributions are centered around the averaged velocities of the individual species. In the resulting momentum balance, individual shear and convected momentum terms are present. Transport coefficients for pure monatomic gases are equal to those from classic theory; for mixtures of such gases new expressions are found that give an excellent description of experimental data. Based on these results, generalized versions of the transport equations are proposed, for dense media and liquids, and limit versions are presented. Important physical parameters are the partial viscosities. The applicability to the experimental situations above is demonstrated. The present theory offers the perspective of evaluating both concentration and velocity profiles, as well as temperature gradients, for individual species in three-dimensional space for molecular transport. Thus it provides a new basis for the modeling of multicomponent transport in a multitude of systems, such as in catalysts, adsorbents, membranes, CVD- and microreactors, but also for the classical problem of the circulation in Stefan tubes. In one of its limits, for long flat or cylindrical channels, it supports the earlier developed velocity profile model (VPM-1) for transport in pores. © 2004 American Institute of Chemical Engineers AIChE J, 51: 79-121, 2005