Exponential shear flows of polymer melts are investigated. These flows are of interest because they share some properties of both planar extension and simple, constant-rate, shear. In particular, embedded points separate exponentially in time, like extensional flows, yet the flow direction and velocity gradient are perpendicular, as for all shear deformations. A comparison between the predictions of the "pom-pom" molecular model of McLeish and Larson [J. Rheol. 42, 81-110 (1998)] in exponential shear and experimental data is presented. The solutions in exponential shear are used to contrast this flow with simple shear and extensional flows and the physical processes driving the solutions are analyzed in each of the three geometries. Accurate quantitative predictions are made using the multimode approach, land the possibility of using exponential shear to obtain the multimode nonlinear spectrum of a melt is explored. It is concluded that although exponential shear is less effective than extensional flows at exposing the influence of the nonlinear parameters of a melt, there are still situations where it is a useful flow for obtaining these parameters. (C) 2001 The Society of Rheology.