  We present results for the relation between a heavy quark mass defined in the
on-shell and $\bar{\rm MS}$ scheme to four-loop order. The method to compute
the four-loop on-shell integral is briefly described and the new results are
used to establish relations between various short-distance masses and the
$\bar{\rm MS}$ quark mass to next-to-next-to-next-to-leading order accuracy.
These relations play an important role in the accurate determination of the
$\bar{\rm MS}$ heavy quark masses.
