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 dependence
are presented whenever available, and formulae valid in the limit of
small bottom mass ($m_{\rm b}^2\ll s$) or of large top mass ($m_{\rm t
}^2 \gg s$) are presented. 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 investigated.
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.