  The distinguishing number $D(G,X)$ of an action of a group $G$ on a set $X$
is the least size of a partition of $X$ such that no element of $G$ acting
nontrivially on $X$ preserves this partition. In this paper we describe the
distinguishing numbers for all actions of the symmetric group $S_n$, for any
$n\geq 3$. This allows us to describe the distinguishing numbers for all graphs
whose automorphism group is isomorphic with a symmetric group. Our description
solves a few open problems posed by various authors in earlier papers on this
topic.
