Josephson junctions translate quantum phase coherence into an electrical response and underpin superconducting sensors and quantum circuits. In conventional junctions, the barrier acts primarily as a passive weak link; however, when the barrier is a quantum material with its own internal degrees of freedom like magnetism, strong correlations, or switchable polarization, the Josephson effect becomes a sensitive probe of symmetry and many-body physics in the interlayer. Here we review progress in "quantum-material Josephson junctions" (QMJJ), focusing on three rapidly advancing barrier families: (1) magnetic barriers, where exchange, noncollinearity, and spin-active scattering enable 0-π-φ ground states, singlet-triplet conversion, and nonreciprocal transport; (2) correlated barriers, where proximity effects acquire many-body character and recent van der Waals Kagome Mott interlayers exhibit field-free Josephson diode behavior; and (3) ferroelectric and multiferroic barriers, where nonvolatile polarization provides an internal control knob and can produce superconducting memory and memristive dynamics. ... Read more