Disclosed is an anisotropic elastomeric nonwoven fibrous web composed of a substantially homogeneous distribution of elastomeric meltblown fibers that are physically aligned along one of the planar dimensions of the web. The anisotropic elastomeric nonwoven web of meltblown fibers has a strength index of at least about 2. Also disclosed is process of making such an anisotropic elastomeric nonwoven web of meltblown fibers as well as a composite elastic material which contains an anisotropic elastomeric nonwoven web of meltblown fibers.