The authors named the rotor having two unequal principal moments of inertia except the polar moment of inertia, "asymmetrical rotor", and studied the vibratory characteristics of the motions of the shaft carrying an asymmetrical rotor analytically and experimentally. The results are as follows : (1) The rotating shaft is statically unstable near its major critical speeds, and the unstable regions become wider as the asymmetry gets larger and the damping forces smaller. (2) The unstable region of the lower major critical speed vanishes as the mounting point of an asymmetrical rotor on the shaft comes to the center of the shaft. (3) There are 8 natural frequencies p_i, p^^-_i (i=1, 2, 3 and 4) in the free vibrations of a four degrees of freedom system treated in this paper, and the relation p^^-_i=2w-p_i always exists. (4) As the asymmetry disappears gradually, the natural frequency p_i approaches that of a symmetrical rotor, and the amplitudes E^^- and F^^- of free vibrations peculiar to p^^-_i disappear. (5) The vibratory characteristics mentioned above (1), (3) and (4) except (2) of the shaft carrying an asymmetrical rotor are the same as those of the flat shaft carrying a symmetrical rotor.