It is shown that for a given number h of benzenoid rings, the polycyclic aromatic hydrocarbons with the maximum number K(max) of Kekule structures (or of perfect matchings) are cata-condensed. The structures of such catafusenes with K(max) present interesting 4- and 3-periodicities in terms of h. By analogy with known catalytic isomerizations of benz[a]anthracene into chrysene, i.e. of a catafusene with a smaller number of perfect matchings into one with a higher number, it was inferred in 1965 that degenerate rearrangements (automerizations) of catafusenes may be possible, and this was demonstrated thermally in 1977 by Scott. Data for the AlHal3-catalyzed automerization of phenanthrene labeled with carbon-13 are reviewed, and the reaction mechanism is discussed. Prospects for new thermal or catalytic isomerizations or automerizations of catafusenes are considered.