A review is made of research on geomagnetic pulsations, which are hydromagnetic waves of natural origin in the frequency range 10$^{-3}$--5 Hz. Methods for diagnostics of the earth's magnetosphere and methods for electromagnetic sounding of the earth's crust on the basis of data from the observation of pulsations are described. It is believed that the electrical properties of the earth's crust are pertinent to the diagnostics of the magnetosphere, while consideration of the spatial structure of the inducing field is useful in geosounding. The fluctuation and critical properties of magnetic storms and pulsations are discussed on the basis of phenomenological models. Estimates of the properties of the magnetosphere and of the interplanetary medium at the earth's orbit can be made more accurate by taking account of the fluctuations in the diagnostic approach of a black box without an input.