Summary: A rigorous mathematical analysis is given for a boundary layer problem for a third-order nonlinear ordinary differential equation which arises in gravity-driven laminar film flow of power-law fluids along vertical walls. Firstly, the problem is transformed into a singular nonlinear two-point boundary value problem of second order. Next, the latter is proved to have a unique positive solution, for which some estimates are established. Finally, the above-mentioned result is turned over to the original problem. The conclusion of this paper is that the boundary layer problem has a unique normal solution if the power-law index is less than or equal to one, and a generalized normal solution if the power-law index is greater than one. Also, the asymptotic behavior of the normal solution at the infinity is displayed.