  We consider the dynamics of gauge-Yukawa theories in the presence of a large
number of matter constituents. We first review the current status for the
renormalization group equations of gauge-fermion theories featuring also
semi-simple groups. In this regime these theories develop an interacting
ultraviolet fixed point that for the semi-simple case leads to a rich phase
diagram. The latter contains a complete asymptotically safe fixed point
repulsive in all couplings. We then add two gauged Weyl fermions belonging to
arbitrary representations of the gauge group and a complex, gauged scalar to
the original gauge-fermion theory allowing for new Yukawa interactions and
quartic scalar self-coupling. Consequently, we determine the leading $1/N_f$
Yukawa and quartic beta functions. Our work elucidates, consolidates and
extends results obtained earlier in the literature. We also acquire relevant
knowledge about the dynamics of gauge-Yukawa theories beyond perturbation
theory. Our findings are applicable to any extension of the standard model
featuring a large number of fermions such as asymptotic safety.
