  The aim of this topical article is to outline the fundamental ideas
underlying the recently developed Fractional Analytic Perturbation Theory
(FAPT) of QCD and present its main calculational tools together with key
applications. For this, it is first necessary to review previous methods to
apply QCD perturbation theory at low spacelike momentum scales, where the
influence of the Landau singularities becomes inevitable. Several concepts are
considered and their limitations are pointed out. The usefulness of FAPT is
discussed in terms of two characteristic hadronic quantities: the
perturbatively calculable part of the pion's electromagnetic form factor in the
spacelike region and the Higgs-boson decay into a $b\bar b$ pair in the
timelike region. In the first case, the focus is on the optimization of the
prediction with respect to the choice of the renormalization scheme and the
dependence on the renormalization and the factorization scales. The second case
serves to show that the application of FAPT to this reaction reaches already at
the four-loop level an accuracy of the order of 1%, avoiding difficulties
inherent in the standard perturbative expansion. The obtained results are
compared with estimates from fixed-order and contour-improved QCD perturbation
theory. Using the brand-new Higgs mass value of about 125 GeV, measured at the
Large Hadron Collider (CERN), a prediction for $\Gamma_{H\to b\bar{b}}=2.4 \pm
0.15 {\rm MeV}$ is extracted.
