  Let $\mathcal{F},\mathcal{G}$ be two cross-intersecting families of
$k$-subsets of $\{1,2,\ldots,n\}$. Let $\mathcal{F}\wedge \mathcal{G}$,
$\mathcal{I}(\mathcal{F},\mathcal{G})$ denote the families of all intersections
$F\cap G$ with $F\in \mathcal{F},G\in \mathcal{G}$, and all distinct
intersections $F\cap G$ with $F\neq G, F\in \mathcal{F},G\in \mathcal{G}$,
respectively. For a fixed $T\subset \{1,2,\ldots,n\}$, let $\mathcal{S}_T$ be
the family of all $k$-subsets of $\{1,2,\ldots,n\}$ containing $T$. In the
present paper, we show that $|\mathcal{F}\wedge \mathcal{G}|$ is maximized when
$\mathcal{F}=\mathcal{G}=\mathcal{S}_{\{1\}}$ for $n\geq 2k^2+8k$, while
surprisingly $|\mathcal{I}(\mathcal{F}, \mathcal{G})|$ is maximized when
$\mathcal{F}=\mathcal{S}_{\{1,2\}}\cup \mathcal{S}_{\{3,4\}}\cup
\mathcal{S}_{\{1,4,5\}}\cup \mathcal{S}_{\{2,3,6\}}$ and
$\mathcal{G}=\mathcal{S}_{\{1,3\}}\cup \mathcal{S}_{\{2,4\}}\cup
\mathcal{S}_{\{1,4,6\}}\cup \mathcal{S}_{\{2,3,5\}}$ for $n\geq 100k^2$. The
maximum number of distinct intersections in a $t$-intersecting family is
determined for $n\geq 3(t+2)^3k^2$ as well.
