We report on recent experimental and theoretical work on the pinch-off of dilute solutions of flexible polymers. Owing to the strong extensional hardening of such solutions, pinch-off of a liquid drop is delayed. Instead, long threads of uniform thickness form, whose radius decreases exponentially in time. We derive a relationship between the thread radius and the extensional viscosity. When the thread radius has decreased to about 10 microns, the thread becomes unstable to the rapid growth of small ``blisters''. Observations are in strong disagreement with conventional models for dilute suspensions. The ensuing dynamics are very rich, and include iterated instabilities and periodic ``breathing''. For sufficiently high polymer concentrations, the fluid never breaks, and a solid nanometer-sized thread is left behind.