Abstract: What happens when a ferromagnet at infinite temperature is suddenly cooled to zero temperature and evolves by spin-flip dynamics? In continuum theories, a coarsening domain mosaic emerges that leads the system to the ground state. However, discrete spin systems exhibit much richer behaviors. For the two-dimensional kinetic Ising model, the ground state is reached about 2/3 of the time, and the time evolution is characterized by two distinct time scales, the longer of which arises from topological defects. There is also a deep connection between domain topologies and continuum percolation. In three dimensions, the ground state is not reached because of topological constraints and the relaxation time grows exponentially with system size. The coarsening of the q-state Potts model is richer still. On the square lattice, spin avalanches may occur at long times that lead to large-scale domain rearrangements. On the triangular lattice and for q=3, a three-hexagon state arises with probability ~0.16 and the relaxation to this state is governed by a time that scales as L^2ln(L).