The thesis examines grain growth in different systems in one and two dimensions. We have used the phase field crystal (PFC) model to perform the simulations. The simulation results are compared to theoretical predictions. In 1D we briefly examine the solid-liquid coexistence state. Then we study the marginal stability analysis of the PFC equation in 1D, for the parabolic and hyperbolic PFC equations. The marginal stability analysis is performed on a solidification process, where a solid front is propagating into a homogeneous liquid. We test the predictions made for the front velocity and the wavenumber forming behind the front. Our results are in agreement with the predictions for the front velocity but not for the wavenumber in the hyperbolic case. We discuss possible causes of this. In 2D we study the dynamics of a two-grain system. We examine the case for different misorientation and temperature, and compare the results to the Cahn-Taylor model. We find what appears to be grain growth stagnation for the colder systems. However, there are strong indications that this could be an effect of the PFC model itself, and not the system.