The steady motion of a Newtonian fluid in a rectangular enclosure open on its upper side is considered under the action of thermocapillary forces due to surface-tension gradients along the free surface. An asymptotic solution, in the limiting case of the aspect ratio A → 0, is found and discussed for the cases where the surface deformation may be neglected, that is for contact angles at the lateral walls equal to ½π and very small values of the crispation number. The flow field has also been analysed in a wide range of the governing parameters A, Mg, Cr, by a computational model particularly appropriate to simulate free-surface flows. For A [Lt ] 1 the numerical results confirm the behaviour predicted by the asymptotic theory, while for A [ges ] 1 several characteristic features of the flow-field structure are emphasized. For increasing Mg, the surface layer under the free surface maintains in the mid-section a constant value, dependent only on A, and decreases together with the thermal boundary-layer thickness near the lateral walls. For increasing A, the motion remains confined in a region near the free surface; hence the overall Nu, starting from the pure conduction value (Nu → 1 as A → 0) inccreases with A, reaching a maximum, to tend again to unity as A → ∞. The surface deformation, at least for very small values of the crispation number, seems to have a negligible influence on the qualitative aspects of the flow-field structure.