Group 3 and lanthanide alkyl complexes supported by PNP-pincer ligands were prepared to investigate the sigma-bond metathesis reaction. These complexes were found to effectively carry out this reaction. Alkylidene complexes were prepared by sigma-bond metathesis with gamma-hydrogens of neopentyl groups to generate metallacyclobutanes which undergo a retro-[2+2] reaction. These complexes were also competent hydrogenation, semihydrogenation, hydrosilation, semihydosilation, and cross-dehydocoupling (amines and silanes) catalysts. The mechanisms of these catalytic reactions as well as the sigma-bond metathesis reaction in general were thoroughly studied by computational techniques. The sigma-bond metathesis reaction was probed by energy decomposition analysis to reveal that polarizability of the metal complex is key to promoting efficient metathesis. A new energy decomposition analysis was developed to treat these bonds in unbiased ways. Work towards the preparation of 1,3-dilithiopropane, a useful molecule for possible preparation of alkylidene complexes, was reported.