Separation processes account for 6% of the annual US energy expenditure, 50% of which is consumed by distillation alone, Therefore, it is not too surprising that distillation, the work horse of the chemical process industry, is under attack by emerging technologies based on membranes and adsorption, whose proponents claim enormous potential savings in energy expenditure. Moreover, the massive scale of use plus the energy intensiveness implies that even small improvements in the efficiency of distillation processes can result in large gains in energy savings. Such improvements can come from developing a fundamental understanding of the fluid mechanics of tray columns, which has heretofore been lacking and is the subject of this paper. The flow on a distillation tray is governed by the equations of mass and momentum conservation in three-dimensions. These equations are reduced here to a set of two-dimensional equations by averaging them across the depth of the fluid film flowing across the tray, The depth-averaged equations are then solved by a Galerkin/finite element technique. The evolution of film height and flow fields are determined for three types of trays that are commonly found in the laboratory and in actual plants : rectangular trays, circular trays, and so-called race track trays. Sample results include development and growth of eddies or zones of recirculation on various types of trays, variation of film height with position on a tray, and effect of tray geometry, flow rate, and physical properties on tray holdup. Occurrence of eddies and large height variations on trays can have detrimental consequences in vapor-liquid contacting operations. Therefore, the new rigorous computations should prove indispensable in developing column designs that avoid or minimize them.