The book is devoted to modern asymptotic methods widely used in nonlinear dynamics and mechanics of a deformable solid. The authors summarized their many years of experience in this area, which was reflected in a large number of articles and monographs, and also took into account the achievements of colleagues. The presentation is based on examples, while the authors tried to avoid complex formal calculations and justifications, preferring to describe the main ideas and algorithms. Considerable attention is paid to summation methods, which are inextricably linked with modern asymptotic approaches.