The book is devoted to vibration mechanics and vibration rheology - new mathematical approaches to studying the effect of vibration on non-linear mechanical systems. These approaches, initiated in Russia at the end of the last century, were developed both in Russia and in other countries. The book reflects the main results obtained. After outlining the basics of these approaches, numerous applications are considered. Attention is paid to peculiar, sometimes paradoxical effects that can be used and are already used in the creation of new machines and technologies.