Metallocene derivatives of the transition metals from groups 3 and 4 undergo catalytic reactions with hydrocarbons. The reactions of C-H bonds by "σ bond metathesis" and some fundamental transformations that comprise chain initiation, propagation, transfer and stereocontrol in α-olefin polymerization with metallocene catalysts have been the focus of research efforts in the Bercaw group over the past fifteen years. Mechanistic studies that accommodate the relative reactivities of C-H bonds in σ bond metathesis and a general mechanism are presented. Permethylscandocene hydride and alkyl derivatives react rapidly and cleanly with ethylene, and the mechanism of olefin insertion and chain transfer by β-H elimination are summarized. Described is the evolution of more active scandium and yttrium catalysts for isospecific propylene polymerization, together with investigations of the mechanisms for stereocontrol. The important features of syndiospecific polymerizations are described, and a new, more active zirconocene catalyst system for producing syndiotactic polypropylene is presented.