In order to provide new experimental facts required for constructing a non-linear constitutive equation for crystalline polymers, some tests were conducted by the use of polyethylene (PE) under various strain histories such as tension, torsion, tension-torsion proportional strain path, tension-torsion cruciform strain path, tension-torsion circular strain path, and various cyclic loadings. It is shown that (i) since the stress-strain behaviour for PE is not very sensitive to hydrostatic pressure, the equivalent stress and strain of von Mises type are useful for its description; (ii) the stress range at a constant strain amplitude at partly reversed cyclic loading tends to increase with an increase in the number of cycles, but the stress amplitude at fully reversed cyclic loading is nearly independent of the number of cycles; (iii) the degree of cyclic softening or hardening is relatively small compared with that of polypropylene; and (iv) under the conditions tested here, the effect of strain history on the stress-strain response is hardly observed for PE.