The present book is devoted to the study of the stability of elastic systems under the action of non-conservative forces. It is wellknown that for such systems the usual methods of the theory of elastic stability, which are based on an examination of forms of equilibrium close to the undisturbed form, are in general no longer applicable. Here we need to use more general methods and more involved means of investigation.The book contains an introduction and four chapters. The first chapter covers general problems, their formulation and methods of solution. It is based on a paper read by the author at the Third All-Soviet Mathematical Conference in Moscow in 1956. The remaining chapters are devoted to applications. The second chapter considers the stabilityof elastic systems under the action of non-conservative forces which during the process of loss of stability behave according to some pre-determined law (so called “follower” forces). The third chapter considers the stability of high-speed rotating elastic rotors under the action of various disturbing forces, for example, forces of internal friction, hydrodynamic and electric forces, etc. The fourth chapter deals with problems of stability of elastic systems in a high-speed gas flow; particular attention is paid to the problem of supersonic flutter of elastic plates and shells. A number of problems are con­ sidered in non-linear form, which enables the behavior of the system to be studied after loss of stability. It will be seen that all these problems are of considerable interest in present day mechanical, aeronautical and rocket engineering.