Let (R, m) be a Noetherian local ring, I an ideal of R and M, N two finitely generated R-modules. The first result of this paper is to prove a vanishing theorem for generalized local cohomology modules which says that H^j_I(M, N) = 0 for all j > dim(R), provided M is of finite projective dimension. Next, we study and give characterizations for the least and the last integer r such that Supp(H^r_I(M, N)) is infinite.