The review summarizes the development of Skyrme's soliton approach to the description of baryon structure. In contrast to existing literature on this subject, the principal attention is devoted not to the pragmatic aspects of the model, but rather to its initial ideas, to its deep topological content and to such subtle problems as the existence of solutions, the attainability of the absolute energy minimum based on the hedgehog ansatz, and so on. It is exactly these features of the Skyrme approach that are, in the authors' opinion, the main advantages compared to other schemes used in strong-interaction physics. The material assembled in the review will, first, enable the reader to gain a deeper understanding of the structure and special features of the Skyrme model and, second, will serve as an adequate base for further modifications and for development of more realistic scenarios of processes in low-energy QCD, the need for which is beyond any doubt.