  A set of permutations $I \subset S_n$ is said to be {\em k-intersecting} if
any two permutations in $I$ agree on at least $k$ points. We show that for any
$k \in \mathbb{N}$, if $n$ is sufficiently large depending on $k$, then the
largest $k$-intersecting subsets of $S_n$ are cosets of stabilizers of $k$
points, proving a conjecture of Deza and Frankl. We also prove a similar result
concerning $k$-cross-intersecting subsets. Our proofs are based on eigenvalue
techniques and the representation theory of the symmetric group.
