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