  We present the phase diagram and associated fixed points for a wide class of
Gauge-Yukawa theories in $d=4+\epsilon$ dimensions. The theories we investigate
involve non-abelian gauge fields, fermions and scalars in the Veneziano-Witten
limit. The analysis is performed in steps, we start with QCD$_d$ and then we
add Yukawa interactions and scalars which we study at next-to- and
next-to-next-to-leading order. Interacting infrared fixed points naturally
emerge in dimensions lower than four while ultraviolet ones appear above four.
We also analyse the stability of the scalar potential for the discovered fixed
points. We argue for a very rich phase diagram in three dimensions while in
dimensions higher than four certain Gauge-Yukawa theories are ultraviolet
complete because of the emergence of an asymptotically safe fixed point.
