  Two families $\mathcal{A}$ and $\mathcal{B}$, of $k$-subsets of an $n$-set,
are {\em cross $t$-intersecting} if for every choice of subsets $A \in
\mathcal{A}$ and $B \in \mathcal{B}$ we have $|A \cap B| \geq t$. We address
the following conjectured cross $t$-intersecting version of the Erd\H
os--Ko--Rado Theorem: For all $n \geq (t+1)(k-t+1)$ the maximum value of
$|\mathcal{A}||\mathcal{B}|$ for two cross $t$-intersecting families
$\mathcal{A}, \mathcal{B} \subset\binom{[n]}{k}$ is $\binom{n-t}{k-t}^2$. We
verify this for all $t \geq 14$ except finitely many $n$ and $k$ for each fixed
$t$. Further, we prove uniqueness and stability results in these cases,
showing, for instance, that the families reaching this bound are unique up to
isomorphism. We also consider a {\em $p$-weight} version of the problem, which
comes from the product measure on the power set of an $n$-set.
