  We classify the weakly interacting fixed points of general gauge theories
coupled to matter and explain how the competition between gauge and matter
fluctuations gives rise to a rich spectrum of high- and low-energy fixed
points. The pivotal role played by Yukawa couplings is emphasized. Necessary
and sufficient conditions for asymptotic safety of gauge theories are also
derived, in conjunction with strict no go theorems. Implications for phase
diagrams of gauge theories and physics beyond the Standard Model are indicated.
