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This review gives an outline of the modern theory of nonlinear oscillations. This field, originated about four decades ago by the late Prof. B. van der Pol, acquired a considerable momentum in recent years with a gradually increasing number of ramifications, some of which are still in a state of development. For this reason the selection of topics for this review is narrowed down to matters which have reached a state of a reasonable codification. It has been assumed that a certain amount of familiarity with these topics is available on the part of the reader so that only essential points are emphasized. Topological methods and analytical methods are written in a condensed manner, while special attention is given to the stroboscopic method, which is probably less known, and which is used extensively later. A survey of the principal nonlinear phenomena investigated under the assumption of near linearity is given. Phenomena of parametric excitation, subharmonic resonance and synchronization are outlined with some detail, but those of the so called asynchronous actions are treated less completely as their detailed presentation would lengthen the paper unduly. The final section of this review deals with the so called "piece-wise linear phenomena;" it transcends already purely analytic methods and belongs to recent developments concerning nonlinear and nonanalytic oscillations often encountered in the theory of automatic regulation and control.