The Earth's magnetic field has two intriguing features ; one is its dipole nature ; and the other is its reversal of the north-south polarity. The purpose of this study is to simulate these features of the geomagnetic field and to understand the physical mechanism behind it by means of large scale computer simulation. The simulation model is as follows : We consider a spherical shell region between two concentric spheres. The temperatures of the spherical boundaries are kept constant and uniform ; the inner sphere is hot and outer sphere is cold. An electrically conducting fluid is confined in this spherical shell region, corresponding to the liquid iron in the Earth's outer core. When the temperature difference is sufficiently large, convection motion of the electrically conducting fluid sets about. The fundamental equations are the magnetohydrodynamic (MHD) equations with time development equations of temperature field. As a result of our computer simulation, spontaneous generation of dipole magnetic field and its intermittent reversals are successfully reproduced. detail analysis of the simulation data indicates that the dynamo-generated magnetic field has two stable states ; one is high energy state and the other is low energy state. One of interesting findings in this simulation is hat the dipole reversals take place only in the high energy state. Another interesting finding is that quadrupole moment grows, exceeding the dipole moment before the reversal. The computational grid system used in our simulation, by which we could successfully reproduced the dipole field generation and reversals, were spherical latitude-longitude grid. When we applied this code to the Earth Simulator, we found that the latitude-longitude grid is not suitable to this vector-parallel architecture. We, therefore, devised a new kind of spherical grid system based on the overset (or Chimera) grid methodology, and named it Yin-Yang grid. The Yin-Yang grid is successfully applied to the spherical shell MHD dynamo simulation.