Hysteretic Josephson junctions have been fabricated by growing c-axis oriented trilayer Bi2Sr2CaCu2O8/Bi2Sr2(Ca,Bi,Sr)7Cu8O20/Bi2Sr2CaCu2O8 single-crystal thin films and patterning them into devices. To our knowledge, these are the first cuprate superconductor junctions to exhibit current-voltage characteristics with substantial hysteresis. Current flow in the devices is in the vertical (c axis) direction, through an ultrathin (approximately 30 angstrom) barrier layer. The current-voltage response shows large, sharp Shapiro steps under microwave illumination.