  We introduce and study mutation of torsion pairs, as a generalization of
mutation of cluster tilting objects, rigid objects and maximal rigid objects.
It is proved that any mutation of a torsion pair is again a torsion pair. A
geometric realization of mutation of torsion pairs in the cluster category of
type $A_n$ or $A_{\infty}$ is given via rotation of Ptolemy diagrams.
