Given a sequence of positive integers , let denote the family of all sequences of positive integers such that for all . Two families of sequences (or vectors), , are said to be -cross-intersecting if no matter how we select and , there are at least distinct indices such that . We determine the maximum value of over all pairs of -cross-intersecting families and characterize the extremal pairs for , provided that . The case is quite different. For this case, we have a conjecture, which we can verify under additional assumptions. Our results generalize and strengthen several previous results by Berge, Borg, Frankl, Furedi, Livingston, Moon, and Tokushige, and answers a question of Zhang.