The humidification-dehumidification desalination system includes a humidifier, a dehumidifier, and a carrier gas that is humidified in the humidifier and condensed in the dehumidifier to produce purified water. The humidifier preferably includes multiple stages in a staircase configuration in which at least one stage has a perforated or porous plate and the carrier gas is bubbled through the bottom of the plate into the “liquid to be purified” atop the plate. The dehumidifier may also include multiple stages in a staircase configuration in which at least one stage has a perforated or porous plate and the carrier gas is bubbled through the bottom of the plate into the purified liquid atop the plate. In both the humidifier and the dehumidifier, the liquid travels downward from one stage to the next stage. The system may be operated in a closed loop, so that the carrier gas is recirculated from the dehumidifier to the humidifier.