Let 2^[n] and (\(\matrix{{\left[ n \right]} \cr i \cr } \)) be the power set and the collection of all i-subsets of {1, 2, …, n}, respectively. We call t (t ≥ 2) families \({{\cal A}_1},{{\cal A}_2}, \ldots ,{{\cal A}_t} \subseteq {2^{\left[ n \right]}}\) cross-intersecting if A_i ∩ A_j ≠ ∅ for any \({A_i} \in {{\cal A}_i}\) and \({A_j} \in {{\cal A}_j}\) with i ≠ j. We show that, for n ≥ k +l, l ≥ r ≥ 1, c > 0 and \({\cal A} \subseteq \left( {\matrix{{\left[ n \right]} \cr k \cr } } \right),{\cal B} \subseteq \left( {\matrix{{\left[ n \right]} \cr l \cr } } \right)\), if \({\cal A}\) and \({\cal B}\) are cross-intersecting and \(\left( {\matrix{{n - r} \cr {l - r} \cr } } \right) \le \left| {\cal B} \right| \le \left( {\matrix{{n - 1} \cr {l - 1} \cr } } \right)\), then

This implies a result of Tokushige and the second author (Theorem 3.1) and also yields that, for n ≥ 2k, if \({{\cal A}_1},{{\cal A}_2}, \ldots ,{{\cal A}_t} \subseteq \left( {\matrix{{\left[ n \right]} \cr k \cr } } \right)\) are non-empty cross-intersecting, then

which generalizes the corresponding result of Hilton and Milner for t = 2. Moreover, the extremal families attaining the two upper bounds above are also characterized.