Known for its surprising properties of precession and oscillation, the simplest and oldest gyroscope design, along with its modifications, remains in use today, continuing to manifest the gyroscopic effects. The inertial torques generated by the rotating masses of the spinning rotor result in the gyroscope motions. The laboratory gyroscope, supported on one side, demonstrates nutation and oscillation with large amplitude, a motion known as galloping. Human beings may not have created all the unique designs of gyroscopic devices, so there are probably more examples of their unusual motions. It is necessary to describe and explain gyroscopic effects, which exhibit these exceptional properties mathematically based on their physical principles. Mathematical models for interconnected inertial torques acting on spinning objects and the dependence of angular velocities of their rotations about axes describe all gyroscopic effects. This Chapter explores analytical solutions for the motion of a well-balanced ordinary top, a top with an eccentric mass, and a gyroscope nutation. The action of interconnected inertial torques of spinning objects and the principle of mechanical energy conservation present the mathematical models for these examples. This dual approach is essential and enables the resolution of all gyroscopic effects for all rotating objects and devices.