Twenty-five years after the discovery of a vast class of organic reactions named "pericyclic reactions" by Woodward and Hoffmann, ab initio quantum mechanics provides a detailed analysis of the geometries, energies, and electronic characteristics of the transition structures of these reactions. Common features are found in all these reactions, and generalizations permit prediction of other transition-structure geometries and energies. At the same time, great diversity is observed-from strongly bonded, rigid, closed-shell entities to weakly interacting, flexible diradical structures.