Publisher Summary This chapter discusses the codification of nonlinear mechanics progressed along a definite theoretical pattern. The development of the theory of linear differential equations in the middle of the last century resulted in a tendency to linearize the nonlinear problems so as to bring them within the scope of the linear theory. This trend resulted in the development of the well-known method of small motions, owing to which gained in the uniformity of treatment of the various problems. The drawback of the procedure is, however, in the restriction of problems. This review deals only with the systems with one degree of freedom and is limited to the case when the parameter is a small number. A greater emphasis is laid on the equations of the first approximation as probably the only ones that are of interest in applications. A brief outline of the theory of the approximations of higher orders is also discussed. Generalizations for systems with several degrees of freedom do not present any particular difficulty except in added complications in calculations.