The importance and actuality of investigation Space Magnetic Traps in the Universe and in Magnetosphere are based on following four factors:1. Space Magnetic Traps in the Universe for Cosmic Ray particles (Astroparticles, Run Away Particles) on different studies of the Universe's evolution and in different objects (galaxies of different types, quasars, nucleus of galaxies, Sun and stars, Heliosphere and Stellar Spheres, solar and stellar winds, magnetospheres of planets, and so on) are very interested and important in the frame of fundamental science (Astrophysics, Plasma Physics and Magneto-Hydrodynamics, Nuclear and Elementary Particle Physics, Geophysics). This problem is interested also for applications (e.g., problem of controlled thermo-nuclear reactions in magneto-plasmas traps as main source for energy in near future for the Earth's Civilization).2. The Magnetosphere is the nearest giant natural laboratory where with satellites and ground measurements is possible to investigate different plasmas and energetic processes in space caused by interaction of high kinetic energy solar wind plasmas with frozen in Interplanetary Magnetic FieldsIMF and its perturbations (Interplanetary Coronal Mass EjectionsICMEs, Interplanetary Shock WavesISWs, Interplanetary Interaction RegionsIIRs, and so on) with the rotated main geomagnetic field. This interaction leads to a dynamic transformation of magnetic fields in the Magnetosphere, formation Space Magnetic Traps (Radiation Belts), generation and trapping of high energy particles (which can be called Magnetospheric Cosmic RaysMCR), and generation of different types of instabilities and electromagnetic radiations. These processes are 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 in the Universe. This research presents an important basis for fundamental space in frame of Astrophysics, Plasma Physics and Magneto-Hydrodynamics, Nuclear and Elementary Particle Physics, Geophysics.3. In modern time, technology, economics, navigation, TV, internet, radio-connections, military aspects, and the life of people on our planet are strongly connected by the work of many satellites moving inside the Magnetosphere. Different processes and MCR in the magnetosphere influence how the satellites work and often lead to satellite malfunctions, sometimes completely destroying their electronics and satellites become 'dead'. The described research can be considered as a basis for developing methods of forecasting dangerous situations for satellites on different orbits and to decrease the risk of satellite malfunctions and of losing them. It means that this research has important practical applications.4. The interaction of ICME, ISW, and IIR with the Magnetosphere leads to the generation of big magnetic storms accompanied by Cosmic Ray Forbushdecrease and precursory effects in Galactic Cosmic Ray (GCR) intensity. These magnetic storms are dangerous, not only for satellites, but also for Earth's surface regarding technology, radio-connections, car accidents, and human health (e.g., increasing frequency of infarct myocardial and brain strokes). Investigation of causes of magnetic storms 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 and space climate influence on the technology, radio-connections, navigation, transportation, and human health on the Earth in dependence of altitude and geomagnetic latitude.