The importance and actuality of the geomagnetospheres research are based on the following three factors: 1. The geomagnetosphere is the nearest giant natural laboratory, where it is possible by multiple satellites and ground measurements to investigate in detail many different plasmas and energetic processes in space; these are caused by the interaction of high kinetic energy solar wind plasmas and their perturbations (Coronal Mass Ejections - CME, Interplanetary Shock Waves ISW, Interplanetary Interaction Regions IIR) frozen in Interplanetary Magnetic Fields (IMF) with the rotated main geomagnetic field. This interaction leads to a dynamic transformation in magnetic fields in the geomagnetosphere, generation and trapping of high energy particles (which are also called Magnetospheric Cosmic Rays MCR), and generation of instabilities and electromagnetic radiations. These processes are in principle similar to processes in magnetospheres of other planets and their moons, in the atmosphere of the Sun and other stars, in interplanetary and in interstellar space, and in many different astrophysical objects. Put simply, this research is an important basis for fundamental space and astrophysical science. 2. In modern times, technology, economics, navigation, television, internet, radio connections, military and all aspects of peoples lives on our planet are strongly connected with the work of many satellites moving inside the geomagnetosphere. Different processes and MCR in the geomagnetosphere influence satellites often lead to satellite malfunctions and sometimes fully destroying them. The described research can be considered as a basis for developing methods of forecasting dangerous situations for satellites in different orbits, and to decrease the risk of satellite malfunction and loss. 3. The interaction of CME, ISW, and IIR with the geomagnetosphere leads to the generation of big magnetic storms accompanied with Forbush decrease and precursory effects in Galactic Cosmic Ray (GCR) intensity. These magnetic storms are dangerous not only for satellites, but also on the Earths surface for technology, radio connections, car accidents, and human health (e.g., increasing frequency of infarct myocardial and strokes). Investigations of magnetic storm causes can help to develop methods of their forecasting and decrease the level of magnetic storm hazards. Therefore, the other practical application of this research is connected with the problem of space weather influence on the technology, radio connections, navigation, transportation, and human health on Earth in regards to altitude and latitude.