  Interacting fixed points in four-dimensional gauge theories coupled to matter
are investigated using perturbation theory up to three loop order. It is shown
how fixed points, scaling exponents, and anomalous dimensions are obtained as a
systematic power series in a small parameter. The underlying ordering principle
is explained and contrasted with conventional perturbation theory and Weyl
consistency conditions. We then determine the conformal window with asymptotic
safety from the complete next-to-next-to-leading order in perturbation theory.
Limits for the conformal window arise due to fixed point mergers, the onset of
strong coupling, or vacuum instability. A consistent picture is uncovered by
comparing various levels of approximation. The theory remains perturbative in
the entire conformal window, with vacuum stability dictating the tightest
constraints. We also speculate about a secondary conformal window at strong
coupling and estimate its lower limit. Implications for model building and
cosmology are indicated.
