QCD Corrections to the $e^+e^-$ Cross Section and the $Z$ Boson Decay Rate

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.