  We classify all finite group actions on knots in the 3-sphere. By
geometrization, all such actions are conjugate to actions by isometries, and so
we may use orthogonal representation theory to describe three cyclic and seven
dihedral families of symmetries. By constructing examples, we prove that all of
the cyclic and four of the dihedral families arise as symmetries of prime
knots. The remaining three dihedral families apply only to composite knots. We
also explain how to distinguish different types of symmetries of knots using
diagrammatic or topological data. Our classification immediately implies a
previous result of Sakuma: a hyperbolic knot cannot be both freely periodic and
amphichiral. In passing, we establish two technical results: one concerning
finite cyclic or dihedral subgroups of isometries of the 3-sphere and another
concerning linking numbers of symmetric knots.
