  We prove that there exists a constant $c_0$ such that for any $t \in
\mathbb{N}$ and any $n\geq c_0 t$, if $A \subset S_n$ is a $t$-intersecting
family of permutations then$|A|\leq (n-t)!$. Furthermore, if $|A|\ge
0.75(n-t)!$ then there exist $i_1,\ldots,i_t$ and $j_1,\ldots,j_t$ such that
$\sigma(i_1)=j_1,\ldots,\sigma(i_t)=j_t$ holds for any $\sigma \in A$. This
shows that the conjectures of Deza and Frankl (1977) and of Cameron (1988) on
$t$-intersecting families of permutations hold for all $t \leq c_0 n$. Our
proof method, based on hypercontractivity for global functions, does not use
the specific structure of permutations, and applies in general to
$t$-intersecting sub-families of `pseudorandom' families in
$\{1,2,\ldots,n\}^n$, like $S_n$.
