A modular building system for constructing a multi-story building includes prefabricated modular units. Each modular unit includes a horizontal slab for forming a floor/ceiling of the building, a joist extending horizontally along a first edge of the slab, the joist extending below the slab and extending horizontally beyond the first edge of the slab to form a receiving lip, columns positioned on top of the slab along the first edge, and connectors installed in cavities formed in the unit. A floor of the multi-story building is formed by arranging the modular units so that the slab of a first modular unit is positioned on the receiving lip of a second modular unit. An additional floor of the multi-story building is formed by arranging the modular units so that the joist portion of an upper modular unit is positioned on the columns of a lower modular unit.