The main definitions and theorems are given, the mathematical apparatus of vibration mechanics is described - a new direction in the theory of mechanical vibrations, characterized by a mathematical approach to the description and study of a wide range of phenomena that occur under the effect of vibration on non-linear mechanical systems and underlying a number of modern machines and technologies. Special sections are devoted to vibration mechanics of mechanisms and machines, synchronization of rotors, vibration movement and displacement, vibration rheology.