We use computational homology to characterize the geometry of complicated time-dependent patterns. Homology provides very basic topological (geometrical) information about the patterns, such as the number of components (pieces) and the number of holes. For three- dimensional patterns it also provides the number of enclosed cavities. We apply these techniques to patterns generated by experiments on spiral defect chaos, as well as to numerically simulated patterns in the Cahn-Hilliard theory of phase separation and on spiral wave patterns in excitable media. Some of the results obtained with these techniques include distinguishing patterns at different parameter values, detecting complicated dynamics through the computation of positive Lyapunov exponents and entropies, comparing experimental and numerically simulated data, and quantifying boundary effects on finite size domains.