The monograph is devoted to a number of poorly explored problems of non-linear electrodynamics, nuclear physics, and general relativity that are very important for astrophysics, geophysics, and cosmology. In it new theoretical methods are proposed to solve the problems under consideration and to explain a number of puzzling experimental and observational data. In the book the authors study the non-linear equations of the Yang-Mills theory and their applications to several unsolved problems of electrodynamics. They find and investigate a non-linear generalisation of the Yukawa meson theory to describe the phenomenon of nuclear saturation and various properties of nuclei. They propose a generalised Dirac's equation to describe the quark structure and anomalous magnetic moments of nucleons. They consider a number of unsolved problems of general relativity and study anomalous properties of gravitational waves. Finally, they find and investigate a generalisation of the Einstein gravitational equations based on Weyl's geometry to explain puzzling properties of spiral galaxies and to apply it to several unsolved problems of cosmology.