  Let $n$, $r$, $k_1,\ldots,k_r$ and $t$ be positive integers with $r\geq 2$,
and $\mathcal{F}_i\ (1\leq i\leq r)$ a family of $k_i$-subsets of an $n$-set
$V$. The families $\mathcal{F}_1,\ \mathcal{F}_2,\ldots,\mathcal{F}_r$ are said
to be $r$-cross $t$-intersecting if $|F_1\cap F_2\cap\cdots\cap F_r|\geq t$ for
all $F_i\in\mathcal{F}_i\ (1\leq i\leq r),$ and said to be non-trivial if
$|\cap_{1\leq i\leq r}\cap_{F\in\mathcal{F}_i}F|<t$. If the $r$-cross
$t$-intersecting families $\mathcal{F}_1,\ldots,\mathcal{F}_r$ satisfy
$\mathcal{F}_1=\cdots=\mathcal{F}_r=\mathcal{F}$, then $\mathcal{F}$ is well
known as $r$-wise $t$-intersecting family. In this paper, we describe the
structure of non-trivial $r$-wise $t$-intersecting families with maximum size,
and give a stability result for these families. We also determine the structure
of non-trivial $2$-cross $t$-intersecting families with maximum product of
their sizes.
