  One of the hallmarks of 6D superconformal field theories (SCFTs) is that on a
partial tensor branch, all known theories resemble quiver gauge theories with
links comprised of 6D conformal matter, a generalization of weakly coupled
hypermultiplets. In this paper we construct 4D quiverlike gauge theories in
which the links are obtained from compactifications of 6D conformal matter on
Riemann surfaces with flavor symmetry fluxes. This includes generalizations of
super QCD with exceptional gauge groups and quarks replaced by 4D conformal
matter. Just as in super QCD, we find evidence for a conformal window as well
as confining gauge group factors depending on the total amount of matter. We
also present F-theory realizations of these field theories via elliptically
fibered Calabi-Yau fourfolds. Gauge groups (and flavor symmetries) come from
7-branes wrapped on surfaces, conformal matter localizes at the intersection of
pairs of 7-branes, and Yukawas between 4D conformal matter localize at points
coming from triple intersections of 7-branes. Quantum corrections can also
modify the classical moduli space of the F-theory model, matching expectations
from effective field theory.
