This book introduces the reader to methods of mathematical modeling and solving non-stationary (dynamic) problems of the theory of magnetoelasticity, as well as to give an idea of the wealth of physical effects caused by the interaction of electromagnetic and mechanical phenomena in magnetically active elastic thin bodies. The studies are mainly limited to a model of isotropic body under the assumption of small deformations. In the first chapter of the book, based on the basic connected nonlinear equations and relations of mechanics and quasi-static electrodynamics of continuum media, a system of equations of magnetoelasticity, surface conditions, and governing equations describing the behavior of disturbances in a magnetoactive medium interacting with an external magnetic field is obtained. On this basis, in Chapters 2 and 3, using the main equations and relations of magnetoelastic vibrations and stability of magnetically soft, thin plates and shells are obtained. By solving specific applied problems, a number of qualitative and quantitative results were identified, caused by the interaction of mechanical and magnetic phenomena in ferromagnetic thin bodies. An approximate formula is obtained to determine magnetohydrodynamic pressure on the oscillating surfaces of plate flowing by supersonic flow of perfectly conducting gas in the presence of magnetic field. This formula is the generalization of well-known formula obtained on the basis of the classical piston theory of gas dynamics in the case of magneto-gas-dynamic flow. On this basis, it became possible to solve complex problems of aeromagnetoelasticity. In the 4th and 5th chapters magnetoelastic processes in superconducting thin shells located in stationary and non-stationary magnetic fields are studied. Two-dimensional equations and corresponding conditions are obtained, which characterize vibrations and stability of superconducting cylindrical and spherical shells under the influence of the given magnetic field. By solving specific problems, the possibility of loss of both static and dynamic stability of thin superconducting bodies under the influence of external magnetic field has been established. The sixth chapter is devoted to mathematical modeling and investigation of issues of dynamics of magnetostrictive plates in magnetic fields (stationary and non-stationary) of several orientations. To study the processes of magnetoelastic interaction in the plate under consideration with complex physical properties of its material, the main postulates of the classical theories and methods were used. The influence of plate in-homogeneity on the processes under consideration was also studied. Dynamic processes in layered plates have been studied. It is shown that heterogeneity is sufficient to control optimally the studied dynamic processes, especially those that arise as a result of interaction. The last 7th chapter is devoted to the investigation of stability of dielectric thin plates in a supersonic flow of perfectly conducting gas in the presence of magnetic field. The problems were studied in both linear and nonlinear formulations. Based on the formula obtained by the authors specific problems of stability were solved. Influence of magnetic field on the flutter characteristics is studied.