Non-Newtonian or viscoelastic fluids do many things which Newtonian fluids cannot do, and many of the most striking examples involve free (deformable) surfaces. I will discuss two new examples from this class of problems: the buckling instability of a stretched funnel-shaped surface, and the dimpled shape of a falling viscoelastic drop. Two different mathematical approaches are taken to understand these observations, involving in one case an elastic membrane model for the stress boundary layer at the free surface, and in the other case a classic perturbation approach, where the small parameter is the nondimensional elastic relaxation time (Deborah number).