In this paper, we establish some left and right multiplicative perturbation theorems concerning local C -semigroups when the generator A of a perturbed local C -semigroup S (⋅) may not be densely defined and the perturbation operator B is a bounded linear operator from D ( A ) ¯ $\overline {D(A)}$ into R ( C ) such that C B = B C on D ( A ) ¯ $\overline {D(A)}$ , which can be applied to obtain some additive perturbation theorems for local C -semigroups in which B is a bounded linear operator from [ D ( A )] into R ( C ) such that C B = B C on D ( A ) ¯ $\overline {D(A)}$ . We also show that the perturbations of a (local) C -semigroup S (⋅) are exponentially bounded (resp., norm continuous, locally Lipschitz continuous, or exponentially Lipschitz continuous) if S (⋅) is.