Persistent homology was introduced in [[ELZ00]] and quickly developed into the most influential method in computational topology. There were independent developments of this idea preceding this paper, and we mention the little known paper by Marston Morse [[Mor40]], the spectral sequences introduced by Jean Leray in [[Ler46]], the notion of prominence in mountaineering [[Mun53]], the size function introduced by Frosini [[Fro90]], and the study of fractal sets by Robins [[Rob99]]. Perhaps the fast algorithm described in [[ELZ00]] triggered the explosion of interest we currently observe because its availability as software facilitates the application to a broad collection of problems and datasets.