Using the theory of elasticity for large deformations and a strain energy function of the formW=½&PHgr;(I1−3)+½&PSgr;(I2−3), where &PHgr; and &PSgr; are constants, formulas for the phase velocity and damping of torsional waves of small amplitude in stretched viscoelastic isotropic vessels of arbitrary wall thickness and filled with a viscous fluid under pressure have been derived and experimentally tested (frequency range 0.3–33 Hz; viscosity of the fluids between 0.01 and 13 P; pressure between 0 and 4×105dyn cm−2; stretches up to twice the original length) in vertically or horizontally suspended latex tubes and in such tubes floating on a pillow of pressurized air. The agreement between theory and experiment is satisfactory. The phase velocity increases with the stretch, but decreases with an increase of the fluid viscosity. The damping decreases with an increase of the stretch. At the higher frequencies the damping increases with an increase of the viscosity of the fluid, but at the lower frequencies the opposite is the case. The influence of pressure both on phase velocity and damping proved small in the pressure range that was considered (0–4×105dyn cm−2).