Abstract Torsional guided waves are often used to detect the defects in a hollow cylinder. To realize the excitation of the torsional guided waves with high efficiency, the transient vibration responses of finite, semi-infinite and infinite hollow cylinders to external torsional forces must be clarified theoretically. In this study, the method of eigenfunction expansion is employed to solve the above problems. The exact analytical solutions derived by this method are not only explicit but also concise. Furthermore, the analytical solution of the transient torsional vibration of the finite hollow cylinder is numerically evaluated. The results obtained agree well with those simulated by the finite element method.