The purpose in writing this book is to give an historical overview of a new challenging field of research, and equip the readers with the mathematical basis of gravitoelectromagnetic theories and their applications to advanced science and technology. The first chapter introduces the historical background of electrogravity, especially on the Biefeld-Brown effect. The second chapter gives several explanations on the Biefeld-Brown effect and other related phenomena, with a concern on the Einstein's Unified Field Theory of Gravitation and electromagnetism and gravitational anomaly induced by the massive electrostatic charges of planets. The third chapter is concerned with the electrogravitic effect related to the zero point energy fluctuation in the vacuum, introduced from the standpoint of quantum electrodynamics. The fourth chapter discusses other electromagnetic gravity control devices including the Heim theory and their applications for space flight. The fifth chapter has shown that the Abraham force is the analogue of the Magnus force, and it thus represents the formation of vortex structures, of electromagnetic nature, in the physical vacuum: the electromagnetotoroid which can generate gravitational field. The sixth chapter deals with the plasma theory of the Universe and the role played by the gravito-electromagnetic forces generated by the plasma permeating the space between planets. And the last chapter shows the application on advanced aviation systems and future prospects of these technologies. This is a textbook written for both researchers and professional scientists, which provides the mathematical basis for readers to introduce the basic concept of gravitoelectromagnetic theories and also discusses their application to advanced science and technologies.