The book deals with Ultra-Low-Frequency (ULF)-electromagnetic waves observed on the Earth and in Space. These are so-called geomagnetic variations or pulsations. Alfvén's discovery related to the influence of the strong magnetic field on the conducting fluids (magnetohydrodynamics) led to development of the concept that the ULF-waves are magnetospheric magnetohydrodynamic (MHD)-waves. MHD-waves at their propagation gather information about the magnetosphere, ionosphere, and the ground. There are two applied aspects based on using the ULF electromagnetic oscillations.The first one is the ground-based diagnostics of the magnetosphere. This is an attempt to monitor in the real time the magnetosphere size, distance to the last closed field-lines, distribution of the cold plasma, etc.The second one is the deep electromagnetic sounding of the Earth. The basis for these studies is the capability of any electromagnetic wave to penetrate a conductor to a finite depth. The ULF-waves can reach the depth of a few hundred kilometers.Thus, geophysicists now have a unique tool that can be applied to study the solid Earth as well as its gas-plasma shells - the ionosphere and magnetosphere.This book integrates topics pertaining to all scales of the MHD-waves, emphasizing the linkages between the ULF-waves below the ionosphere on the ground and magnetospheric MHD-waves. It will be most helpful to graduate and post-graduate students, familiar with advanced calculus, who study the science of MHD-waves in the magnetosphere and ionosphere.