In general the migration of grain boundaries during grain growth in polycrystalline materials seems quite easy to describe. At elevated temperatures small grains show a tendency to shrink, while their larger neighbors grow, effectively reducing the total grain boundary surface area and therewith also reducing the total free energy of the microstructure. However, the true nature of this phenomenon is much more complex. It is a constant struggle between the global topological requirement of a three dimensional space filling grain network and the local requirement of equilibrium at the grain boundary junctions. Then again, understanding and controlling the polycrystalline grain microstructures of solid materials is absolutely essential to improving their properties, such as toughness, strength and diffusivity. To this end, much effort has been put into investigations of grain microstructures and their temporal evolution during recrystallization and grain growth by experiments as well as by analytical theories in the last 50 years. About 30 years ago researchers started to model these two phenomena using a variety of computer simulation methods, e.g., the Monte Carlo Potts model, phase-field algorithms, Surface Evolver, and vertex method. Those mesoscopic simulations have the advantage that they give insights regarding the temporal development of the microstructures as observed during the simulation. Hence they can be understood as in-situ computer experiments. In particular, the Monte Carlo Potts model has the advantage that it is in its basics rather simple but in its specifics quite complex allowing for example the investigation of normal grain growth as well as junction controlled grain growth in nanocrystalline materials. In the first years, two dimensional computer models were dominating, because of the complexity, the size of computational memory, and the time-expense of carrying out three dimensional simulations. However, especially in the past decade the advances in computational hardware as well as in numerical algorithms have enabled researchers to investigate complex polycrystalline grain microstructures and their temporal evolution during recrystallization and grain growth as they occur in reality, namely in three dimensions.