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 a mismatching of the QCD running coupling and its corresponding coefficients at each order, leading to well-known renormalization scheme-and-scale ambiguities. However, these ambiguities are unnecessary,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 initial renormalization scale.The elimination of the scale and scheme ambiguities relies heavily on how precisely we know the analytic form of the QCD running coupling.Conventional schemes for defining the QCD coupling suffer from a complex and scheme-dependent renormalization group equation (RGE), which is usually solved perturbatively at high orders due to the entanglement of the scheme-running and scale-running behaviors.In this paper, we show that these complications can be avoided by using the newly suggested $C$-scheme coupling, whose scheme-and-scale running behaviors are governed by the same scheme-independent RGE....Given a measurement which sets the magnitude of the coupling at a specific scale such as $M_Z$,the resulting pQCD predictions, after applying the single-scale Principle of Maximum Conformality (PMC), become independent of the choice of the renormalization scheme and the initial renormalization scale, thus satisfying all conditions of the RGI....We illustrate these features for the non-singlet Adler function and $\tau $ decay to $\nu +$ hadrons at four-loop order. The PMC thus eliminates a serious systematic scheme-and-scale error in pQCD predictions, greatly improving the precision of tests of the Standard Model and the sensitivity of collider experiments to new physics. Show Partial Abstract