Numerical and asymptotic methods are used to study the surface-tension driven instability of the jet. For the Newtonian case, we establish a relationship between the initial shape of the jet and the asymptotic approach to breakup. For the Oldroyd-B fluid, no breakup occurs, and we study the evolution of the jet for large time. On the other hand, the numerical solutions show finite time breakup for the Giesekus model. For the upper convected Maxwell model, nonunique discontinuous solutions exist, and the choice of the physically appropriate solution depends on regularizing terms in the equations. We discuss one such regularization involving higher order derivatives resulting from taking into account the axial curvature of the free surface.