The lubrication theory is extended for the flow of viscoelastic fluids of the Oldroyd-B type inside thin cavities. The formulation accounts for nonlinearities stemming from inertia effects in the momentum conservation equation, and the upper-convected terms in the constitutive equation. The theory is applied to transient free-surface-flow problems inside a thin (two-dimensional) channel. The influence of fluid elasticity (Deborah number) and retardation on the shape and evolution of the front is examined. It is found that the mean position of the front is dictated by a nonlinear equation of second order. The multiple-scale method is applied to obtain an approximate solution at small Deborah number. Given the existence of a singularity in the limit De --> 0, regular perturbation theory cannot be applied. Comparison with exact (numerical) solution indicates a wide range of validity for the multiple-scale results, which is not necessarily restricted to weakly elastic flows. (C) 2000 Elsevier Science B.V. All rights reserved.