  A technique called analytic perturbation theory, which respects the required
analytic properties, consistent with causality, is applied to the definition of
the running coupling in the timelike region, to the description of inclusive
$\tau$-decay, to deep-inelastic scattering sum rules, and to the investigation
of the renormalization scheme ambiguity. It is shown that in the region of a
few GeV the results are rather different from those obtained in the ordinary
perturbative description and are practically renormalization scheme
independent.
