  We describe a calculation of the on-shell renormalization factors in QCD and
QED at the three loop level. Explicit results for the fermion mass
renormalization factor Zm and the on-shell fermion wave function
renormalization constant Z2 are given. We find that at O(alpha_s^3) the wave
function renormalization constant Z2 in QCD becomes gauge dependent also in the
on-shell scheme, thereby disproving the ``gauge-independence'' conjecture based
on an earlier two-loop result. As a byproduct, we derive an O(alpha_s^3)
contribution to the anomalous dimension of the heavy quark field in HQET.
