Over the last thirty years an abundance of papers have been writ- ten on adaptive dynamic control systems. Nevertheless, now it may be predicted with confidence that the adaptive mechanics, a new division, new line of inquiry in one of the violently developing fields of cybernetic mechanics, is emerging. The birth process falls far short of being com- pleted. There appear new problems and methods of their solution in the framework of adaptive nonlinear dynamics. Therefore, the present work cannot be treated as a certain polished, brought-to-perfection school textbook. More likely, this is an attempt to show a number of well- known scientific results in the parametric synthesis of nonlinear systems (this, strictly speaking, accounts for the availability of many reviews), as well as to bring to notice author s developments on this question undoubtedly modern and topical. The nonlinear, and practically La- grangian, systems cover a wide class of classical objects in theoretical mechanics, and primarily solid-body (robotic, gyroscopic, rocket-cosmic, and other) systems. And what is rather important, they have a direct trend to practical application. To indicate this discussion, I should like to notice that it does not touch upon the questions concerned with the linear and stochastic con- trolobjects. Investigated are only nonlinear deterministic systems being in the conditions when some system parameters are either unknown or beyond the reach of measurement, or they execute an unknown limited and fairly smooth drift in time.