We give a short presentation of the first genuine QCD five-loop
calculations, viz. analytical contributions of order \alpha_s^4 n_f^2 and
m_q^2 \alpha_s^4 n_f^2 to the vacuum polarization function of
vector currents. These corrections form an important gauge-invariant
subset of the full O(\alpha_s^4) correction to the total cross-section
of e^+ e^- annihilation into hadrons.