The Higgs-boson decay rates into $bb$ and into $gg$ have been evaluated in N4LO, corresponding to order $α4s$ for $bb$ and order $α6s$ for $gg$ final states. After inclusion of the four-loop term, nice stabilization of the series is observed. In a similar context the predictions for the $τ−$ and the $Z-$ decay rate, as well as the $R-$ ratio measured in electron-positron annihilation are presented in order $α4s$. Similar methods are employed for the evaluation of the beta function which governs the running of the quark-gluon coupling in quantum chromodynamics. The five-loop term of this fundamental quantity has been evaluated and the result has quickly been confirmed and even extended to a general gauge group. This five-loop term leads to a further reduction of the theory uncertainty in $αs$, evaluated at the $Z-$ boson or Higgs-boson scale, if originally extracted from $τ-$ lepton decays and subsequently evolved to $mZ$ or $mZ$.