Let α be an ideal of a commutative Noetherian ring R, and let M and N be finitely generated R-modules. Let fα(N) = min{j ≥ 0|Hjα(N) not finitely generated} be the α-finiteness dimension of N. In this paper, among other things, we show that for each i ≤ fα(N), (i) the set of associated prime ideals of generalized local cohomology module Hiα(M, N) is finite, and (ii) Hiα(M, N) is α-cofinite if and only if HOα(HomR(M, Hiα(N))) is so. Moreover, we show that whenever α is a principal ideal, then Hnα(M, N) is α-cofinite for all n.