The authors present an approximation theory for thin filaments, fibers or jets which yields families of transient 1-D models (time-dependent, one-dimensional, closed systems of PDEs). The spatial reduction from three dimensions to one is achieved by axisymmetry together with a local expansion in the radial jet coordinate; this reduction is in contrast to the predominant viscoelastic models in the literature which average out the radial dimension and thereby require moment equations for the computation of higher order corrections. The authors also allow torsional flow effects, which are usually ignored, in a general Johnson–Segalman constitutive law. A formal perturbation theory, based on a slenderness parameter and a compatible velocity-pressure-stress ansatz, is then constructed for the full 3-D free surface boundary problem. This formalism contains all 1-D transient models that govern slender axisymmetric flows of inviscid, viscous, or viscoelastic fluids; specific models follow by positing the dominant...