The difference of the geomagnetic field from the field of the idealized dipole not only creates certain difficulties for studying the geomagnetically trapped radiation but also presents new possibilities for studying the processes which control the particle movement in the magnetosphere. It will be remembered that the fact that the geomagnetic field is not a dipole played a decisive role in the very beginning of the formation of our ideas about the nature of the intense fluxes of the penetrating radiation detected during the satellite flights. In fact, the longitude dependence of the lower boundary of the penetrating radiation in the region of low latitudes according to the properties of the geomagnetic field made it possible to conclude immediately that the detected radiation was caused by the geomagnetically trapped particles. As to the difficulties in the trapped radiation study caused by the existence of the anomalies, they, first of all, result in the original and, at the first glance, incomprehensible form of the particle intensity contours drawn in geographical coordinates for low altitudes. However, when turning to the McIlwain coordinate system, which is naturally connected with the real magnetic field, the seeming chaos in the radiation distribution disappears being unexpectedly replaced by harmony and order. But even in this coordinate system some ambiguities are observed connected with the distortion of the adiabatic invariants during the time comparable with the period of the particle drift around the earth. The thorough analysis of these effects in combination with the observations of the radiation at high altitudes may answer such a principle question as the question about the velocity of the movements of the mirror points. It should be noted that some information has been obtained to date on the movement of the electron-mirror points only and similar information about protons is absent. It is possible that new aspects of the use of the magnetic anomalies for studying the geomagnetosphere will appear in due time.