Rhombohedral-stacked transition-metal dichalcogenides (3R-TMDs), which are distinct from their hexagonal counterparts, exhibit higher carrier mobility, sliding ferroelectricity, and coherently enhanced nonlinear optical responses. However, surface epitaxial growth of large multilayer 3R-TMD single crystals is difficult. We report an interfacial epitaxy methodology for their growth of several compositions, including molybdenum disulfide (MoS), molybdenum diselenide, tungsten disulfide, tungsten diselenide, niobium disulfide, niobium diselenide, and molybdenum sulfoselenide. Feeding of metals and chalcogens continuously to the interface between a single-crystal Ni substrate and grown layers ensured consistent 3R stacking sequence and controlled thickness from a few to 15,000 layers. Comprehensive characterizations confirmed the large-scale uniformity, high crystallinity, and phase purity of these films. The as-grown 3R-MoS exhibited room-temperature mobilities up to 155 and 190 square centimeters per volt second for bi- and trilayers, respectively. Optical difference frequency generation with thick 3R-MoS showed markedly enhanced nonlinear response under a quasi-phase matching condition (five orders of magnitude greater than monolayers).