Let R be a commutative Noetherian ring, I an ideal of R and M an R-module. We wish to investigate the relation between vanishing, finiteness, Artinianness, minimaxness and C-minimaxness of local cohomology modules. We show that if M is a minimax R-module, then the local-global principle is valid for minimaxness of local cohomology modules. Moreover, if Hi I (M) is minimax for all i ≥ n ≥ 1, then Hi I (M) is Artinian for i ≥ n. It is shown that if M is a C-minimax module over a local ring such that Hi I (M) are C-minimax modules for all i < n (respectively i ≥ n), where n ≥ 1, then they must be minimax. Consequently, a vanishing theorem is proved for local cohomology m
