1. The motion of the torque-free gyrostat initially at rest is a permanent rotation about a certain axis.

2. The free gyrostat is investigated as a dynamical model of the dual-spin satellite or the satellite with control wheels.

3. The attitude motion of an asymmetric gyrostat under the self - excited control torque is investigated.

4. The global characters of the permanent rotations of a free gyrostat is analysed in present paper.

5. In this paper the motion of a torque - free asymmetric gyrostat with a mass - eccentric rotor is discussed.