The fundamental study of the propagation of radio waves has formed an important part of the programme of the Radio Research Board, which was established under the Department of Scientific and Industrial Research in 1920. The object of the present paper is to survey the progress made in this field during the decade ending in 1946, and to indicate the manner in which the investigations concerned have been developed to meet the requirements of those responsible for applying radio waves to various practical uses. The researches described cover the radio-frequency spectrum from about 10 kc/s up to 50000 Mc/s or more. They deal with the transmission of radio waves along the surface of the earth, and illustrate the manner in which this transmission is influenced by the electrical properties of the ground and the curvature of the earth and by the corresponding properties and slopes of natural and man-made protuberances on the ground. The characteristics of the ionosphere have been studied intensively for over twenty years, and during the past decade considerable advance has been made in our knowledge of this region of the atmosphere and of the manner in which radio waves are propagated through it. With the advance of radio technique and developments to shorter and shorter wavelengths, it has become clear that wave propagation in the lower atmosphere is greatly affected by the vertical gradient of the temperature and humidity of the air; the associated phenomena have been studied directly, and, in addition, the electrical properties of the constituents of the atmosphere, solid, liquid and gaseous, have been determined at the very high radio frequencies relevant to this portion of the investigation. Finally, the important part played by radio noise of either atmospheric or extraterrestrial origin has been studied with a view to understanding more clearly the manner and circumstances in which such noise can either influence the successful transmission of intelligence by radio waves, or provide a means of increasing our knowledge of such phenomena as those associated with thunderstorms and activity on the surface of the sun. This survey attempts to show what an important part the study of wave propagation plays in numerous applications of radio technique, and demonstrates that there is still much further work to be done before the full requirements of those using the technique can be met in a satisfactory manner.