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The conventional approach to fixed-order perturbative QCD predictions isbased on an arbitrary choice of the renormalization scale, together with anarbitrary range. This {\it ad hoc} assignment of the renormalization scalecauses the coefficients of the QCD running coupling at each perturbative orderto be strongly dependent on the choice of both the renormalization scale andthe renormalization scheme. However, such ambiguities are not necessary, sinceas a basic requirement of renormalization group invariance (RGI), any physicalobservable must be independent of the choice of both the renormalization schemeand the renormalization scale. In fact, if one uses the {\it Principle ofMaximum Conformality} (PMC) to fix the renormalization scale, the coefficientsof the pQCD series match the series of conformal theory, and they are thusscheme independent. It has been found that the elimination of the scale andscheme ambiguities at all orders relies heavily on how precisely we know theanalytic form of the QCD running coupling $\alpha_s$. In this review, wesummarize the known properties of the QCD running coupling and its recentprogresses, especially for its behavior within the asymptotic region. We alsosummarize the current progress on the PMC and some of its typical applications,showing to what degree the conventional renormalization scheme-and-scaleambiguities can be eliminated after applying the PMC. We also compare the PAapproach for the conventional scale-dependent pQCD series and the PMCscale-independent conformal series. We observe that by using the conformalseries, the PA approach can provide a more reliable estimate of the magnitudeof the uncalculated terms. And if the conformal series for an observable hasbeen calculated up to $n_{\rm th}$-order level, then the $[N/M]=[0/n-1]$-typePA series provides an important estimate for the higher-order terms.

The QCD Renormalization Group Equation and the Elimination of Fixed-Order Scheme-and-Scale Ambiguities Using the Principle of Maximum Conformality