The subject of the paper is classical and quantum versions of transition state theory (TST), an theoretical and computational approach to analyzing chemical reactions (but applied also in other areas) known also under names ’activated complex theory’ and ’theory of absolute reaction rates’. The paper, on the basis of recent results in analytical and computational techniques, realizes Wigner’s dynamical picture of transition state theory in phase space for systems with three or more degrees of freedom. To set this in a proper context, the authors first present the historical background and setting of TST.