The book contains an unconventional presentation of the course of the theory of elasticity, based on special sections of the theory of partial differential equations and mathematical analysis. The first chapter in a rather compact form gives a concise presentation of those mathematical disciplines that are already successfully used and can be used in the future when solving various problems of the theory of elasticity at the modern level. The next two chapters are devoted to a concentrated, but at the same time quite complete presentation of the subject of the theory of elasticity, including such relatively new sections as electromagnetic elasticity and brittle fracture mechanics, setting boundary value problems, as well as presenting some techniques for reducing the boundary value problems of the theory of elasticity to classical problems of mathematical physics. In the remaining chapters of the book (chapters VI-VIII), specific mathematical methods indicated in the title are applied to solving certain classes of problems in the theory of elasticity. In some cases, the effectiveness of a particular method is demonstrated by examples of such problems, the solution of which has been obtained only recently. Much attention is paid to both the issues of strict mathematical justification of certain algorithms, and the methods of their numerical implementation.