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Electromagnetic waves at extremely low frequency (ELF) have a remarkable ability to propagate with very low attenuation in the space between the earth's surface and the ionosphere. The resulting fields also are able to propagate to moderately great depths in the ocean in spite of the high conductivity of sea water. The principal drawback in the use of such signals for communication is the inherent inefficiency of the transmitting antenna. But indeed the waves do penetrate to deeply submerged submarines when all other methods fail. This has been the principal motivation of the U.S. Navy's controversial Project Sanguine that envisages a huge antenna facility that could cover an area of 100 by 100 km. This project, now called Seafarer, has stimulated a great deal of research on how ELF waves (in the range from about 1- 1000 Hz) propagate. We review here some of the fascinating developments in this field. Actually, the early investigations of Nicola Tesla have a striking similarity to Project Sanguine. Tesla's prophetic lectures and articles provide an interesting historical perspective to the later developments that have spanned over 70 years. After sketching some of the early history of the subject, we outline the theoretical and physical bases of the propagation phenomena. While the basic concept of the earth-ionosphere waveguide is relatively simple, the physics of the interaction with the ionospheric plasma is complicated, particularly during disturbed conditions.