  Given a sequence of positive integers $p = (p_1, . . ., p_n)$, let $S_p$
denote the family of all sequences of positive integers $x = (x_1,...,x_n)$
such that $x_i \le p_i$ for all $i$. Two families of sequences (or vectors),
$A,B \subseteq S_p$, are said to be $r$-cross-intersecting if no matter how we
select $x \in A$ and $y \in B$, there are at least $r$ distinct indices $i$
such that $x_i = y_i$. We determine the maximum value of $|A|\cdot|B|$ over all
pairs of $r$- cross-intersecting families and characterize the extremal pairs
for $r \ge 1$, provided that $\min p_i >r+1$. The case $\min p_i \le r+1$ 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, Frankl, F\"uredi, Livingston, Moon, and Tokushige, and
answers a question of Zhang.
