Fixed points in quantum and statistical field theory are responsible for
a large variety of phenomena. Low energy fixed points often entail quantum,
thermal, or topological phase transitions, (chiral) symmetry breaking,
or the generation of mass, while high energy fixed points are at the root of
a fundamental definition of QFT such as in asymptotic freedom or safety.

In this talk, I discuss the prospects for UV and IR fixed points in general
4d quantum field theories. This covers an overview of rigorous results at
weak coupling including strict no-go theorems and concrete examples. I also
explain the central role of  non-abelian gauge fields, and how strongly
interacting fixed points can be identified with the help of supersymmetry.
Links with conformal field theory and new directions for model building
are indicated. If time permits, I'll also report on the status for fixed
points in quantum gravity.