Premixed flames provide a unique experimental setting for studying the dynamics of pattern formation. The hotter regions appear as cells and the darker, cooler regions separating the cells appear as cusps and folds. The system has hundreds of states ranging from highly ordered patterns to intermittent and chaotic. Ordered states, consisting of concentric rings of approximately equal-size cells, are observed over a wide range of parameters in the experiments using heavy hydrocarbon-air mixtures on a circular porous plug burner at low pressure. Ratcheting states result from secondary bifurcations of ordered states in which symmetry is broken by small spatial inhomogeneities. In the ratcheting states, one or more rings drift (~1 deg/sec) in a circular path, speeding up and slowing down in a characteristic manner. Rotating states are traveling waves this translationally invariant system. The origin of these states is well-understood, and similar have been observed in a number of fluid systems. Hopping states are typically observed in isobutane-air cellular flames at parameter values between those corresponding to two consecutive ordered states. These states are characterized by a hopping motion in which individual cells abruptly change their angular position. Intermittent states are observed whenever an ordered pattern appears at irregular intervals of time. Two types of intermittent states have been observed in the experiments: heteroclinic connections and the unstable 2-cell state. Disordered states are observed at high flow rates. These states have no discernible ring structure.