  The hadronic vacuum polarization correction to the $g$ factor of a bound
electron is investigated theoretically. An effective hadronic Uehling potential
obtained from measured cross sections of $e^- e^+$ annihilation into hadrons is
employed to calculate $g$ factor corrections for low-lying hydrogenic levels.
Analytical Dirac-Coulomb wave functions, as well as bound wave functions
accounting for the finite nuclear radius are used. Closed formulas for the $g$
factor shift in case of a point-like nucleus are derived. In heavy ions, such
effects are found to be much larger than for the free-electron $g$ factor.
