  A family of permutations (\mathcal{A} \subset S_{n}) is said to be
(t)-\textit{intersecting} if any two permutations in (\mathcal{A}) agree on at
least (t) points, i.e. for any (\sigma, \pi \in \mathcal{A}), (|\{i \in [n]:
\sigma(i)=\pi(i)\}| \geq t). It was recently proved by Friedgut, Pilpel and the
author that for (n) sufficiently large depending on (t), a (t)-intersecting
family (\mathcal{A} \subset S_{n}) has size at most ((n-t)!), with equality
only if (\mathcal{A}) is a coset of the stabilizer of (t) points (or
`(t)-coset' for short), proving a conjecture of Deza and Frankl. Here, we first
obtain a rough stability result for (t)-intersecting families of permutations,
namely that for any (t \in \mathbb{N}) and any positive constant (c), if
(\mathcal{A} \subset S_{n}) is a (t)-intersecting family of permutations of
size at least (c(n-t)!), then there exists a (t)-coset containing all but at
most a (O(1/n))-fraction of (\mathcal{A}). We use this to prove an exact
stability result: for (n) sufficiently large depending on (t), if (\mathcal{A}
\subset S_{n}) is a (t)-intersecting family which is not contained within a
(t)-coset, then (\mathcal{A}) is at most as large as the family \mathcal{D} & =
& \{\sigma \in S_{n}: \sigma(i)=i \forall i \leq t, \sigma(j)=j \textrm{for
some} j > t+1\} && \cup \{(1 t+1),(2 t+1),...,(t t+1)\} which has size
((1-1/e+o(1))(n-t)!). Moreover, if (\mathcal{A}) is the same size as
(\mathcal{D}) then it must be a `double translate' of (\mathcal{D}), meaning
that there exist (\pi,\tau \in S_{n}) such that (\mathcal{A}=\pi \mathcal{D}
\tau). We also obtain an analogous result for (t)-intersecting families in the
alternating group (A_{n}).
