In a new rheometer for polymer melts equibiaxial and planar elongations were performed at 150 degreesC with a low and a high density polyethylene (LDPE and HDPE, respectively), and equibiaxial elongations at 170 degreesC with a commercial polystsyrene (PS); for simple elongations a previously described rheometer was used. Each sample is clamped by metal belts that introduce the elongation and measure the forces that act upon each clamp. The samples are small and are supported by inert gas. By particle tracking and video recording the deformation history and the test performance are determined. The resulting elongational viscosities mu(i)(t) (i = u, e, p for the different elongational modes) are newly defined such that at small strain rates they are equal to the linear viscoelastic shear viscosity, etadegrees(t). The ratio mu(i)/etadegrees represents the nonlinear behavior: mu(i)/etadegrees > 1 means strain hardening and mu(i)/etadegrees < 1 strain softening. Hardening occurs in simple and in planar elongation in the flow direction, more for LDPE than for HDPE and is smallest for PS (simple elongation only). In equibiaxial elongation, there is softening followed by much (LDPE), little (HDPE), and no (PS) hardening. Planar elongation causes softening in the cross direction (more for HDPE than for LDPE). At the end of planar elongation, the samples become hazy and form holes. (C) 2003 The Society of Rheology.