In systems with many local degrees of freedom, high-symmetry points in the phase diagram can provide an important starting point for the investigation of the broader phase diagram. In systems with both spin and orbital (or valley) degrees of freedom, SU(4)-symmetric models can serve as such a starting point. In this talk I will discuss SU(4) quantum antiferromagnets on the triangular lattice that arise from Mott-insulating phases of fermions with four flavors. I will consider different fillings of the SU(4) fermions, which lead to different representations of SU(4) on each site, and present our studies aiming to uncover the nature of the ground state in these systems.