The one-dimensional, gravity-driven film flow of a linear (l) or exponential (e) Phan-Thien and Tanner (PTT) liquid, flowing either on the outer or on the inner surface of a vertical cylinder or over a planar wall, is analyzed. Numerical solution of the governing equations is generally possible. Analytical solutions are derived only for: (1) 1-PTT model in cylindrical and planar geometries in the absence of solvent, beta (eta) over bar (s)/((eta) over bar (s) + (eta) over bar (p)) = 0, where (eta) over bar (p) and (eta) over bar (s) are the zero-shear polymer and solvent viscosities, respectively, and the affinity parameter set at xi = 0; (2) l-PTT or e-PTT model in a planar geometry when beta = 0 and xi not equal 0; (3) e-PTT model in planar geometry when beta = 0 and xi = 0. The effect of fluid properties, cylinder radius, (R) over tilde, and flow rate on the velocity profile, the stress components, and the film thickness, (R) over tilde, is determined. On the other hand, the relevant dimensionless numbers, which are the Deborah, De = (lambda) over tilde(U) over tilde/(H) over tilde, and Stokes, St = (rho) over tilde(g) over tilde(H) over tilde (2) /(eta) over tilde (p) + (eta) over tilde (s) (U) over tilde, numbers, depend on (H) over tilde and the average film velocity, (U) over tilde. This makes necessary a trial and error procedure to obtain (H) over tilde a posteriori. We find that increasing De,., or the extensibility parameter epsilon increases shear thinning resulting in a smaller St. The Stokes number decreases as (R) over tilde/(H) over tilde decreases down to zero for a film on the outer cylindrical surface, while it asymptotes to very large values when (R) over tilde/(H) over tilde decreases down to unity for a film on the inner surface. When.xi not equal 0, an upper limit in De exists above which a solution cannot be computed. This critical value increases with epsilon and decreases with xi.