Actually MEMS is a widely spread and well known term - Micro-Electro-Mechanical Systems. MEMS gyroscope is well known almost twenty years. For the time being NEMS is still a new term and it is meant Nano-Electro-Mechanical Systems. It is discussed NEMS gyroscope as a new angular rate sensor. It is suggested to use recently invented narrowest electrical wire, just three atoms wide or graphene nanoribbons as a new type of the angular rate sensor. The basic principle of the new device is similar to the Sagnac effect for the light speed. It is described the method of angular rate determination. The voltage difference between two wire coils with different line coupling can be expressed in applied rotation (angular) rate and velocity of electric current. The obtained scale factor depends on the magnitude of the current, number of the coil turns, the coil's radius, the cross-section area of the wire and specific (unit) resistance of the wire. The device can be produced from hybrid metal-organic chalcogenide nanowires or graphene nanoribbons. It is proposed any ways of sensor improvement.