Abstract: A polycrystal is an agglomerate of single crystal grains into a single dense solid. Grain boundaries (GBs) are the surfaces across which crystal orientation changes discontinuously and hence separate grains from each other. Since the lowest energy state of an ensemble of atoms at low temperature is a single crystal, GBs are 2d defects with a positive energy per unit area. A polycrystal will evolve over time via grain boundary migration such that the average grain size increases in time, lowering the total energy. There is a thermodynamic driving force for GB migration proportional to the product of the GB energy and the mean curvature of the boundary. This leads to curvature flow and normal grain growth. Von Neumann and Mullins gave an exact topological relation for the evolution of grain size in 2d and MacPherson and Srolovitz gave exact relations that can be applied in all dimensions. Although such exact relations are known, a theoretical understanding of grain growth remains elusive for several reasons. First, unlike in the exact analyses, grain boundary energies are not isotropic but depend on the relative orientation of the grains they divide and on the orientation of the grain boundary normal. The first effect gives rise to crystallographic texture and the second to grain boundary faceting. I will discuss how these modify the earlier exact results. Second, while the theories describe the evolution of grain size for fixed topology, grain topology changes as grain faces and edges (triple junctions) disappear. I will discuss the evolution of topology during grain growth. Third, the theories assume that three-grain boundaries meet along (triple) lines and that the angles at which they meet are fixed. Experiment and simulations show that this is not the case and, in fact, dissipation in the system gives rise to velocity-dependent triple junction angles. In all of these cases, I will present theoretical arguments and simulations that help elucidate these phenomena.