The motion and shape of a drop of dilute polymer solution translating through a quiescent viscous Newtonian fluid is investigated. Experimentally, the drop may exhibit a stable dimple at its trailing edge. At higher volumes the dimple extends far into the interior of the drop, and pinches off via a Rayleigh-type instability, injecting droplets of the bulk fluid into the drop. At even larger volumes, a toroidal shape develops. A perturbation analysis is used to reproduce the dimpled shape and to calculate the drag on the drop.