Let R be a noetherian ring, alpha an ideal of R such that dim R/alpha = 1 and M a finite R-module. We will study cofiniteness and some other properties of the local cohomology modules H-alpha(i)(M). For an arbitrary ideal alpha and an R-module M (not necessarily finite), we will characterize alpha-cofinite artinian local cohomology modules. Certain sets of coassociated primes of top local cohomology modules over local rings are characterized.