  Let \fa be an ideal of a commutative Noetherian ring R and M and N two
finitely generated R-modules. Let \cd_{\fa}(M,N) denote the supremum of the i's
such that H^i_{\fa}(M,N)\neq 0. First, by using the theory of Gorenstein
homological dimensions, we obtain several upper bounds for \cd_{\fa}(M,N).
Next, over a Cohen-Macaulay local ring (R,\fm), we show that
\cd_{\fm}(M,N)=\dim R-\grade(\Ann_RN,M), provided that either projective
dimension of M or injective dimension of N is finite. Finally, over such rings,
we establish an analogue of the Hartshorne-Lichtenbaum Vanishing Theorem in the
context of generalized local cohomology modules.
