sigma-Bond activations of R1-R2 and R1-X1 (R1, R2 = H, alkyl, aromatics, etc.; X1 = electronegative group) by transition-metal complexes are classified into two main categories: sigma-bond activation by a metal (M) center and that by a metal-ligand bond. The former is classified into two subcategories: concerted oxidative addition to M and stepwise oxidative addition via nucleophilic attack of M. The latter is also classified into two subcategories: heterolytic activaton by M-X2 (X2 = anion ligand) and oxidative addition to M-L (L = neutral ligand). In the concerted oxidative addition, charge transfer (CT) occurs from the M d orbital to the sigma* antibonding orbital of R1-R2, the clear evidence of which is presented here. The concerted oxidative additions of Ph-CN, Me-CN, and Ph-Cl to a nickel(0) complex are discussed as examples. The stepwise oxidative addition occurs through nudeophilic attack of M to R1-X1 to form an ion-pair intermediate. In the nucleophilic attack, CT occurs from the M d(sigma) to either the sigma* orbital or empty p(pi) orbital of R1-X1 Solvation plays a crucial role in stabilizing the transition state and ion-pair intermediate. The oxidative addition reactions of Ph-I, CH3-Br, and Br2B(OSiH3) to platinum(0), platinum(II), and palladium(0) complexes are discussed. In the heterolytic activation of R1-R2 by an M-X2 bond, R1 and R2 are bound with M and X2, respectively, indicating that RI becomes anion-like and R2 becomes cation-like. CT mainly occurs from the X2 ligand to the sigma* antibonding orbital of R1-R2 and also from R1 to the M empty d orbital. In the oxidative addition to an M-L moiety, RI is bound with M, R2 is bound with L, and thus-formed L-R2 is bound with M. The oxidative addition reaction of the Si H bond of silane to Cp2Zr(C2H4) and that of the H-H bond of H-2 to Ni[MesB(o-Ph2PC6H4)(2)] are discussed as examples. The importance of the sigma-bond activation in such catalytic reactions as nickel(0)-catalyzed phenylcyanation of alkyne, nickel(0)-catalyzed carboxylation of phenyl chloride, ruthenium(II)-catalyzed hydrogenation of carbon dioxide, and the Hiyama cross-coupling reaction is discussed based on theoretical studies.