  We discuss the St\"uckelberg-Peterman extended renormalization group
equations in perturbative QCD, which express the invariance of physical
observables under renormalization-scale and scheme-parameter transformations.
We introduce a universal coupling function that covers all possible choices of
scale and scheme. Any perturbative series in QCD is shown to be equivalent to a
particular point in this function. This function can be computed from a set of
first-order differential equations involving the extended beta functions. We
propose the use of these evolution equations instead of perturbative series for
numerical evaluation of physical observables. This formalism is free of
scale-scheme ambiguity and allows a reliable error analysis of higher-order
corrections. It also provides a precise definition for $\Lambda_{\overline{\rm
MS}}$ as the pole in the associated 't Hooft scheme. A concrete application to
$R(e^+e^- \to {\rm hadrons})$ is presented.
