How polymer molecules uncoil and stretch under the influence of an elongational or stretching flow has been the subject of considerable study and discussion. Many experimental studies have been conducted to try to achieve such flows for dilute polymer solutions, but the recent development of filament stretching devices has made it possible to examine the flows unequivocally. Furthermore, these devices allow examination of the processes by which the polymer molecules relax upon cessation of the flow. Here, we present the predictions of the finitely extensible nonlinear elastic chain model (under the Peterlin approximation) for both the stretching and the relaxation. Calculations of conformational, rheological, and optical properties are given. It is found that the process by which the molecules relax, which is governed primarily by segment relaxation and Brownian motion, is quite different from that by which the molecules stretch. Whereas the stretching process passes through several distinct stages, the relaxation process is much more slow and occurs more uniformly. The predictions agree, at least qualitatively, with experimental results.