The partial decay rate \mbox{$\Gamma(Z\rightarrow b\bar{b})$} of
the $Z$-boson is significantly affected by
higher order QCD corrections.
Of particular interest is the influence of the non-\ vanishing bottom quark
mass. Radiative corrections to the quadratic mass term
of the partial $Z$-width are computed in perturbative QCD.
The calculation of the third order
${\cal O}(\alpha_s^3 m_b^2/M_Z^2)$ contribution for the
vector induced rate and of the second order
${\cal O}(\alpha_s^2 m_b^2/M_Z^2)$ correction
for the axial induced rate
is described. The numerical influence of the results and the theoretical
uncertainties are discussed.