  QCD corrections to the electron positron annihilation cross-section into
hadrons and to the hadronic $Z$ boson decay rate are reviewed. Formal
developments are introduced in a form particularly suited for practical
applications. These include the operator product expansion, the heavy mass
expansion, the decoupling of heavy quarks and matching conditions. Exact
results for the quark mass depen- dence are presented whenever available, and
formulae valid in the limit of small bottom mass ($m_{b}^2\ll s$) or of large
top mass ($m_{t}^2\gg s$) are presen- ted. The differences between vector and
axial vector induced rates as well the classification of singlet and nonsinglet
rates are discussed. Handy formulae for all contributions are collected and
their numerical relevance is investi- gated. Prescriptions for the separation
of the total rate into partial rates are formulated. The applicability of the
results in the low energy region, relevant for measurements around 10 GeV and
below, is investigated and numerical predictions are collected for this energy
region.
