Ultraviolet (UV) fixed points are key for a fundamental definition
of quantum field theory, and the existence of free UV fixed points,
asymptotic freedom, has been known since long. The more recent
discovery that UV fixed points can also be interacting, known as
asymptotic safety, has opened up new territory to look for UV-completions
of the Standard Model, beyond asymptotic freedom or effective theories.
In this talk, I study weakly interacting UV fixed points of gauge theories
with matter. After reviewing earlier findings, I report new results at
4-loop order in perturbation theory. Emphasis is put on the size of the
conformal window, and whether the fixed point dissappears due to a merger,
the loss of vacuum stability, or other. I will also discuss our 
methodology,
which enables a systematic search, and give an outlook into 5-loop effects.

Based on https://arxiv.org/pdf/2307.08747.pdf and forthcoming.