When a viscoelastic fluid blob is stretched out into a thin horizontal filament, it sags gradually under its own weight, forming a catenary-like structure that evolves dynamically. If the ends are brought together rapidly after stretching, the sagging filament tends to straighten by hoisting itself. These two effects have characteristic signatures of the elastic nature of the fluid which set it apart from the behavior of a purely viscous filament analyzed previously [J. Teichman and L. Mahadevan, J. Fluid Mech., 478, 71 (2003)]. Starting from the bulk equations for the motion of a viscoelastic fluid, we use perturbation theory to derive a simplified equation for the dynamics of a viscoelastic filament and analyze these equations in some simple settings to explain our observations.