This book is concerned with the ionosphere and the magnetosphere, and the theory of their effect on radio waves. It includes accounts of some mathematical topics now widely used in this study, particularly W. K. B. approximations, Airy integral functions and integration by steepest descents. The subject is divided into ray theory and full wave theory. Ray theory is useful for high frequencies when the ionosphere is treated as a horizonally stratified medium. The discussion of the magnetosphere, whose structure is more complicated, includes an account of whistlers and ion cyclotron whistlers. The book has been planned both for final year undergraduates and as a reference book for research. It is suitable as a course book on radio propagation for students of physics or electrical engineering or mathematics. Some of the topics are presented from an elementary viewpoint so as to help undergraduates new to the subject. The later parts are more advanced. Because the subject is so large and has seen many important recent advances, some topics have had to be treated briefly, but there is a full bibliography with about 600 references.