  The distinguishing number of a graph $G$ is the smallest $k$ such that $G$
admits a $k$-colouring for which the only colour-preserving automorphism of $G$
is the identity. We determine the distinguishing number of finite $4$-valent
vertex-transitive graphs. We show that, apart from one infinite family and
finitely many examples, they all have distinguishing number $2$.
