A set of permutations on $n$ points is intersecting if, for any two of its elements, there is some point which is sent to the same point by both of them. How large can such a set be? Similarly, a set of partial permutations (meaning injections defined on some $r$ points of the $n$-set, for some fixed $r$) is intersecting if, for any two of its elements, there is some point on which they are both defined and is sent to the same point by both of them. Again, how large can such a set be? We shall survey and discuss some results on these problems. We will also mention some fascinating conjectures in this area. This talk includes joint work with Peter Cameron and Imre Leader.