  The three-loop vacuum polarization function $\Pi(q^2)$ induced by a massive
quark is calculated. A comprehensive description of the method is presented.
From the imaginary part the ${\cal O}(\alpha_s^2)$ result for the production of
heavy quarks $R(s)=\sigma(e^+e^-\to \mbox{hadrons})/\sigma(e^+e^-\to
\mu^+\mu^-)$ can be extracted. Explicit formulae separated into the different
colour factors are given.
