The grain growth in single-phase systems with various anisotropic boundary energies is simulated using a modified two-dimensional (2D) Monte Carlo algorithm. Structural self-similarity both in the grain shape distributions and the grain boundary configurations is found regardless of the degree and type of anisotropy. Parabolic evolution kinetics is observed in most cases. The growth rate represented by the time rate of average grain area change is found to be a function of the grain boundary properties. In some cases, the normalized grain size distributions are not time-invariant, and therefore, current grain growth theories cannot explain conclusively the occurrence of the parabolic kinetics in these cases. The effect of anisotropy on the grain growth kinetics is discussed.