Publisher Summary This chapter deals with the transfer of an atom—usually oxygen, occasionally sulfur—from one species to another. The catalytic capabilities of rhenium compounds burst on the scene about one decade ago, featuring MeReO3 as a catalyst for reactions of hydrogen peroxide. It was quickly verified that peroxorhenium(VII) compounds were the active intermediates. With them, practical reactions and fundamental questions of mechanism could then be resolved. A new generation of oxorhenium compounds has now been prepared. They catalyze oxidation reactions of a different type, and appear to function by a different mechanism. They are oxorhenium(V) compounds that form usually metastable dioxorhenium(VII) intermediates. The mechanisms feature ReV(O)-to-ReVII(O)2 interconversions and catalyze oxygen-atom transfer reactions. The mechanisms show certain diversity as to the steps that enter in a kinetic sense.