The paper gives a review of the research performed at the department of theoretical and applied mechanics for many years and devoted to solving the generalized Chebyshev problems. By a generalized Chebyshev problem is meant the problem in which the solution of a system of motion equations should simultaneously satisfy an additional system of highorder n ≥ 3 differential equations. Therefore, a new class of control problems is introduced positions can be named as an example of such devices. For solving a generalized Chebyshev problem it is offered to apply two theories of motion of nonholonomic systems with high-order constraints developed at the department of theoretical and applied mechanics of Saint Petersburg State University. In this case an additional system of differential equations is considered as a set of high-order programming constraints, the reaction forces of which prove to be the required control forces solving a generalized Chebyshev problem. The first theory is based on constructing a compatib