The ¯rst part of this paper is concerned with the Artinianness of certain local cohomology modules Hia (M) when M is a Matlis re°exive module over a commutative Noetherian complete local ring R and a is an ideal of R. Also, we characterize the set of attached prime ideals of Hn a (M), where n is the dimension of M. The second part is concerned with the vanishing of local cohomology and generalized local cohomology modules. In fact, when R is an arbitrary commutative Noetherian ring, M is an R-module and a is an ideal of R, we obtain some lower and upper bounds for the cohomological dimension of M with respect to a.
