The computational study of pericyclic reactions, an important general class of organic reactions, now provides information about the transition structures of these processes with chemical accuracy, as judged by comparisons with experimental data, such as activation energies, substituent effects on rates, and kinetic isotope effects. This article introduces the methods used to study these reactions and describes how computational results have contributed to the understanding of transition states and mechanisms of the electrocyclic ring openings of cyclobutenes, Diels-Alder cycloaddition reactions, and [3,3]-sigmatropic shifts such as the Cope rearrangement.