Abstract: Two fascinating reactions under study in the early 1970's whose mechanisms were unknown were the alkene metathesis reaction and the alkyne metathesis reaction, in which carbon-carbon double bonds and triple bonds, respectively, are "cleaved" to give all possible products of each type. These reactions provided the impetus (in part) for the discovery and development of new high oxidation state complexes that contain a Metal-C double or triple bonds, so-called alkylidene and alkylidyne complexes, respectively. The lecture will trace the development of this "high oxidation state" organometallic chemistry from 1974 to the present and the development of "well-defined" catalysts for alkene and alkyne metathesis. Applications of this chemistry (among others) include Ring Opening Metathesis Polymerization (ROMP), polymerization of terminal alkynes and dipropargyl malonates to yield polyenes, step-growth polymerization of dienes, and asymmetric reactions for the enantioselective synthesis of organic molecules. The lecture will attempt to show why the Nobel Prize in Chemistry in 2005 was awarded "for the development of the metathesis method in organic synthesis."